#7156202

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

1921 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)(9,10,15,31,16,19,14,20)(11,21,30,17,24,34,25,35)(12,26,13,18,28,39,27,37)(22,32,29,40,23,36,38,33);(2,5)(3,7)(9,11)(10,17)(12,23)(13,29)(14,30)(15,25)(16,24)(18,36)(19,35)(20,34)(21,31)(22,28)(26,33)(27,38)(32,37)(39,40);(1,2)(3,6)(4,5)(7,8)(9,12,15,13,16,28,14,27)(10,18,20,26,19,37,31,39)(11,22,25,38,24,23,30,29)(17,32,34,40,35,36,21,33);(2,5)(6,8)(9,13,15,28,16,27,14,12)(10,18,31,39,19,37,20,26)(11,23,30,38,24,22,25,29)(17,33,34,32,35,40,21,36);(1,3,4,7)(2,6,5,8)(10,19)(11,24)(18,37)(20,31)(22,23)(25,30)(26,39)(29,38)

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=fi,e^2=hi,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, 7063398);