#506414

order	512
nilpotency class	4
isoclinism	isoclinic class of #506414
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#28259
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1747 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

591 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 591 conjugacy classes of subgroups — too many to draw. Only its 136 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)(3,8)(4,19)(5,11)(6,9)(7,26)(10,21)(12,25)(13,14)(15,30)(16,28)(17,32)(18,47)(20,48)(22,49)(23,38)(24,34)(27,43)(29,46)(31,39)(33,54)(35,41)(36,53)(37,62)(40,51)(42,58)(44,60)(45,59)(50,55)(52,57)(56,63)(61,64);(1,3,14,39,5,15,7,18)(2,8,26,47,11,30,13,31)(4,20,23,42,21,53,40,16)(6,17,41,45,24,43,25,46)(9,32,12,29,34,27,35,59)(10,36,38,58,19,48,51,28)(22,52,55,63,54,64,62,44)(33,61,50,56,49,57,37,60);(2,9)(3,15)(6,25)(7,14)(8,27)(10,33)(11,34)(12,13)(16,42)(17,45)(19,49)(22,55)(23,40)(24,41)(26,35)(28,56)(29,47)(30,32)(31,59)(36,61)(37,38)(43,46)(44,64)(48,57)(50,51)(52,63)(54,62)(58,60);(1,4,5,21)(2,10,11,19)(3,16)(6,22,24,54)(7,23,14,40)(8,28)(9,33,34,49)(12,37,35,50)(13,38,26,51)(15,42)(17,44)(18,20)(25,55,41,62)(27,56)(29,57)(30,58)(31,36)(32,60)(39,53)(43,63)(45,64)(46,52)(47,48)(59,61)

Polycyclic presentation:

2^9|b^2=egh,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,f^c=fh,h^c=hi,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413610);