#381332

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1001 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)(11,22,25,32,26,16,24,33)(12,18,27,35,28,36,29,17)(23,37,38,30,39,31,40,34);(1,2)(3,6)(4,5)(7,8)(9,11,15,26)(10,16,19,22)(12,23,28,39)(13,25,14,24)(17,34,35,30)(18,31,36,37)(20,33,21,32)(27,40,29,38);(2,5)(6,8)(9,12)(10,17,19,35)(11,23)(13,27)(14,29)(15,28)(16,30,22,34)(18,21,36,20)(24,38)(25,40)(26,39)(31,33,37,32);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,23)(13,29)(14,27)(15,28)(16,31)(17,20)(19,36)(21,35)(22,37)(24,40)(25,38)(26,39)(30,32)(33,34)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=e,d^2=eg,f^2=i,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,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, 288528);