#369163

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1808 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fgi,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276359);