#7156194

order	512
nilpotency class	3
isoclinism	isoclinic class of #7021351
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=f,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh

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, 7063390);