#501118

order	512
nilpotency class	4
isoclinism	isoclinic class of #501114
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24404
commutator, Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#23873	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

1054 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1054 conjugacy classes of subgroups — too many to draw. Only its 138 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)(5,25)(6,12,30,43)(7,13)(8,14)(9,18,34,27)(11,26,42,29)(16,47)(17,33)(19,44)(20,54)(21,46)(22,38,28,59)(23,39)(24,40)(31,60)(32,48,61,52)(35,51,45,53)(36,55,63,57)(37,56,41,58)(49,64)(50,62);(1,3)(2,9)(4,20)(5,17)(6,27)(7,31)(8,19)(10,36)(11,33)(12,44)(13,45)(14,34)(15,43)(16,23)(18,30)(21,49)(22,57)(24,50)(25,35)(26,53)(28,55)(29,51)(32,39)(37,64)(38,62)(40,63)(41,47)(42,60)(46,61)(48,56)(52,58)(54,59);(1,4,8,24)(2,10,14,40)(3,16,19,49)(5,23,7,21)(6,28,30,22)(9,32,34,61)(11,41,42,37)(12,38,43,59)(13,46,25,39)(15,47,44,64)(17,50,31,20)(18,52,27,48)(26,56,29,58)(33,62,60,54)(35,63,45,36)(51,55,53,57);(1,5,8,7)(2,11,14,42)(3,17,19,31)(4,21,24,23)(6,29,30,26)(9,33,34,60)(10,37,40,41)(12,13,43,25)(15,35,44,45)(16,20,49,50)(18,53,27,51)(22,58,28,56)(32,54,61,62)(36,64,63,47)(38,39,59,46)(48,57,52,55)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341754, #72341943
maximal normal subgroup of	#72341754, #72341943

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408314);