#494120

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

584 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)(9,10,14,22,15,17,29,33,16,21,30,37,13,20,12,19)(11,23,27,38,28,34,25,39,24,18,31,36,26,32,40,35);(2,5)(6,8)(10,17,21,20)(11,24)(12,29)(14,30)(18,34,23,32)(19,37,33,22)(26,28)(35,38,39,36);(9,11)(10,18)(12,25)(13,26)(14,31)(15,28)(16,24)(17,32)(19,35)(20,34)(21,23)(22,38)(27,30)(29,40)(33,39)(36,37);(1,3,4,7)(2,6,5,8)(9,12,13,30,16,29,15,14)(10,19,20,37,21,33,17,22)(11,25,26,27,24,40,28,31)(18,35,34,36,23,39,32,38)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237213, #72239205, #72239683, …
maximal normal subgroup of	#72237213, #72239205, #72239683, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401316);