#501800

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	#24266	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

722 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)(7,9)(8,10)(11,12)(13,14,16,20)(15,23,25,19)(17,21,27,34)(18,22,28,35)(24,39,41,32)(26,38,31,33)(29,36,40,44)(30,43,42,37);(2,4)(5,7,8,11)(6,9,10,12)(15,24)(17,29)(18,28)(19,32)(21,36)(22,35)(23,39)(25,41)(26,42)(27,40)(30,31)(33,43)(34,44)(37,38);(6,10)(9,12)(13,15,16,25)(14,19,20,23)(17,30,27,42)(18,31,28,26)(21,37,34,43)(22,38,35,33)(24,40,41,29)(32,44,39,36);(2,4)(15,25)(19,23)(24,41)(26,31)(30,42)(32,39)(33,38)(37,43)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,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, 408996);