#369091

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1914 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:

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

Polycyclic presentation:

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

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, 276287);