#495968

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
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	C₂⁴	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

584 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)(7,12)(8,10)(9,11)(13,14,18,23,19,16,28,30,17,21,31,36,32,29,22,37)(15,20,26,35,27,25,41,42,24,34,39,44,40,38,33,43);(5,7,9,8)(6,10,11,12)(13,15)(14,20)(16,25)(17,24)(18,26)(19,27)(21,34)(22,33)(23,35)(28,41)(29,38)(30,42)(31,39)(32,40)(36,44)(37,43);(2,4)(14,21)(15,24)(16,29)(23,36)(26,39)(27,40)(30,37)(33,41);(5,8,9,7)(6,10,11,12)(13,16,18,30,19,21,28,36,17,29,31,37,32,14,22,23)(15,25,26,42,27,34,41,44,24,38,39,43,40,20,33,35)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403164);