#493218

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₁₆	C₂³
commutator	C₂²	C₂²×C₄×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂³	C₂³×C₈
prime radical	C₂²×D₄	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

846 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=f,f^2=h,h^2=i,b^a=be,c^a=ci

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400414);