#381335

order	512
nilpotency class	3
isoclinism	isoclinic class of #214664
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1018 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=egi,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^b=hi,d^c=de,f^c=fi,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, 288531);