#495910

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	C₂²×D₄	C₂×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

599 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)(7,10)(8,9)(11,12)(13,14,17,22,18,16,28,29,19,23,31,37,32,30,21,36)(15,20,26,35,27,25,41,42,24,33,39,43,40,38,34,44);(6,9)(10,12)(13,15)(14,20)(16,25)(17,26)(18,27)(19,24)(21,34)(22,35)(23,33)(28,41)(29,42)(30,38)(31,39)(32,40)(36,44)(37,43);(2,4)(5,7,8,11)(6,10,9,12)(15,24)(20,33)(25,38)(26,39)(27,40)(34,41)(35,43)(42,44);(13,16,17,29,18,23,28,37,19,30,31,36,32,14,21,22)(15,25,26,42,27,33,41,43,24,38,39,44,40,20,34,35)

Polycyclic presentation:

2^9|a^2=g,c^2=e,d^2=g,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=ci

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403106);