#503422

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3283	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

995 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=efh,d^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=di

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, 410618);