#393535

order	512
nilpotency class	3
isoclinism	isoclinic class of #196320
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1572 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)(9,10,13,18,14,17,19,32)(11,15,22,29,21,30,26,33)(12,23,25,34,24,16,27,31)(20,28,36,39,35,38,37,40);(9,11,19,22,14,21,13,26)(10,15,18,33,17,30,32,29)(12,20,25,37,24,35,27,36)(16,28,34,39,23,38,31,40);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,20)(13,27)(14,24)(15,28)(17,23)(18,34)(19,25)(21,35)(22,37)(26,36)(29,40)(30,38)(31,32)(33,39)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gi,e^b=ei,g^b=gi,h^b=hi,e^d=ei,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300731);