#492919

order	512
nilpotency class	2
isoclinism	isoclinic class of #20338
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₄²
commutator	C₄	C₂²×C₄×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

778 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,e^2=i,f^2=h,h^2=i,b^a=be,g^a=gi,f^b=fi,d^c=di

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237211, #72239196, #72239206, …
maximal normal subgroup of	#72237211, #72239196, #72239206, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400115);