#493254

order	512
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

612 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

(5,6)(7,11)(8,12)(9,10)(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);(1,2,4,3)(6,10)(11,12)(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);(1,3,4,2)(5,7,9,8)(6,11,10,12)(15,24)(20,33)(25,38)(26,39)(27,40)(34,41)(35,43)(42,44);(5,8,9,7)(6,12,10,11)(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=f,b^2=g,c^2=eg,d^2=ef,f^2=h,h^2=i,b^a=be,d^a=di,c^b=ci

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400450);