#7156226

order	512
nilpotency class	3
isoclinism	isoclinic class of #7021363
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

1914 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)(9,10,16,21,15,22,14,20)(11,23,26,17,27,37,25,34)(12,28,13,19,29,35,30,18)(24,36,39,33,38,32,31,40);(2,5)(3,7)(9,11)(10,17)(12,24)(13,31)(14,26)(15,27)(16,25)(18,32)(19,36)(20,37)(21,23)(22,34)(28,33)(29,38)(30,39)(35,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,30,15,29,16,13)(10,18,21,28,22,19,20,35)(11,24,26,39,27,38,25,31)(17,32,23,33,34,36,37,40);(2,5)(6,8)(9,13,16,29,15,30,14,12)(10,19,21,35,22,18,20,28)(11,24,26,39,27,38,25,31)(17,33,37,32,34,40,23,36);(1,3,4,7)(2,6,5,8)(12,29)(13,30)(18,19)(24,38)(28,35)(31,39)(32,36)(33,40)

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=fi,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,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7063422);