#360068

order	512
nilpotency class	3
isoclinism	isoclinic class of #196278
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1465 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:

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

Polycyclic presentation:

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

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, 267264);