#501550

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

575 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2,3,4)(5,6,8,9)(7,11,12,10)(13,14,19,24)(15,20,28,36)(16,21,18,23)(17,22,31,39)(25,33,42,44)(26,34,32,40)(27,35,41,43)(29,37,30,38);(6,9)(10,11)(13,15)(14,20)(16,26)(17,25)(18,32)(19,28)(21,34)(22,33)(23,40)(24,36)(27,30)(29,41)(31,42)(35,38)(37,43)(39,44);(2,4)(15,25)(16,18)(17,30)(20,33)(21,23)(22,38)(26,27)(28,42)(29,31)(32,41)(34,35)(36,44)(37,39)(40,43);(5,7,8,12)(6,10,9,11)(13,16,19,18)(14,21,24,23)(15,26,28,32)(17,29,31,30)(20,34,36,40)(22,37,39,38)(25,41,42,27)(33,43,44,35)

Polycyclic presentation:

2^9|a^2=efi,d^2=ei,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,d^c=di,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408746);