#499829

order	512
nilpotency class	4
isoclinism	isoclinic class of #499825
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#23900
commutator, Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#60402	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1834 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1834 conjugacy classes of subgroups — too many to draw. Only its 148 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,15)(4,10)(5,13)(6,12)(7,29)(8,14)(9,18)(11,25)(16,48)(17,32)(19,46)(20,55)(21,38)(22,37)(23,44)(24,39)(26,59)(27,42)(28,41)(30,47)(31,53)(33,61)(34,54)(35,56)(36,57)(40,58)(43,52)(45,50)(49,63)(51,62)(60,64);(1,3)(2,9)(4,20)(5,17)(6,18)(7,30)(8,19)(10,35)(11,32)(12,15)(13,43)(14,34)(16,23)(21,50)(22,56)(24,51)(25,52)(26,60)(27,54)(28,61)(29,33)(31,38)(36,45)(37,55)(39,64)(40,48)(41,47)(42,46)(44,63)(49,58)(53,57)(59,62);(1,4,8,24)(2,10,14,39)(3,16,19,50)(5,23,7,21)(6,26,27,22)(9,31,34,63)(11,40,41,36)(12,37,42,59)(13,44,29,38)(15,45,46,48)(17,51,30,20)(18,53,54,49)(25,57,28,58)(32,55,47,62)(33,64,43,35)(52,56,61,60);(1,5,8,7)(2,11,14,41)(3,17,19,30)(4,21,24,23)(6,25,27,28)(9,32,34,47)(10,36,39,40)(12,13,42,29)(15,43,46,33)(16,20,50,51)(18,52,54,61)(22,57,26,58)(31,62,63,55)(35,45,64,48)(37,38,59,44)(49,56,53,60)

Polycyclic presentation:

2^9|c^2=i,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,h^a=hi,c^b=cg,f^b=fh,g^b=gi,d^c=di,e^c=eh,g^c=gi,f^d=fhi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281882, #72282072, #72282134, …
maximal normal subgroup of	#72281882, #72282072, #72282134, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407025);