#361204

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

930 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)(3,7)(4,6)(5,8)(9,10,13,19,14,20,29,18)(11,16,22,15,24,31,23,32)(12,25,27,17,28,35,40,36)(21,37,38,33,26,30,39,34);(1,3,4,8)(2,5,6,7)(9,11,13,22,14,24,29,23)(10,15,18,16,20,32,19,31)(12,21,27,38,28,26,40,39)(17,30,25,33,36,37,35,34);(2,6)(5,7)(9,11)(10,16)(12,26)(13,22)(14,24)(15,19)(17,34)(18,32)(20,31)(21,28)(23,29)(25,30)(27,39)(33,36)(35,37)(38,40);(2,6)(5,7)(9,12,13,27,14,28,29,40)(10,17,18,25,20,36,19,35)(11,21,22,38,24,26,23,39)(15,30,16,33,32,37,31,34)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268400);