#367666

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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	#2358
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1083 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)(9,10,16,20,15,21,14,22)(11,23,25,17,13,33,26,19)(12,27,32,38,31,18,30,37)(24,35,28,36,29,39,40,34);(9,11)(10,17)(12,28,31,40)(13,15)(14,26)(16,25)(18,35,27,39)(19,21)(20,33)(22,23)(24,32,29,30)(34,37,36,38);(2,5)(6,8)(9,12)(10,18)(11,24)(13,29)(14,30)(15,31)(16,32)(17,34)(19,36)(20,37)(21,27)(22,38)(23,39)(25,28)(26,40)(33,35);(1,3,4,7)(2,6,5,8)(9,13,15,11)(10,19,21,17)(12,29,31,24)(14,25,16,26)(18,36,27,34)(20,23,22,33)(28,32,40,30)(35,37,39,38)

Polycyclic presentation:

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

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, 274862);