#393465

order	512
nilpotency class	3
isoclinism	isoclinic class of #32898
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1364 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)(9,10,13,17,14,18,24,30)(11,15,21,27,20,26,34,38)(12,16,23,29,22,28,35,39)(19,31,33,40,32,25,36,37);(9,11,13,21,14,20,24,34)(10,15,17,27,18,26,30,38)(12,19,23,33,22,32,35,36)(16,25,29,37,28,31,39,40);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,23)(14,22)(15,25)(17,29)(18,28)(20,32)(21,33)(24,35)(26,31)(27,37)(30,39)(34,36)(38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dg,g^b=gi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300661);