#381129

order	512
nilpotency class	3
isoclinism	isoclinic class of #214688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1113 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 288325);