#7145826

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996395
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

2060 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)(9,10,14,29,15,19,13,18)(11,20,22,16,24,31,23,32)(12,17,27,39,28,35,26,34)(21,33,36,30,37,40,25,38);(2,5)(3,7)(9,11)(10,16)(12,25)(13,22)(14,23)(15,24)(17,33)(18,31)(19,32)(20,29)(21,26)(27,37)(28,36)(30,34)(35,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,17)(13,18)(14,29)(15,19)(20,23)(21,30)(22,31)(24,32)(25,33)(26,34)(27,39)(28,35)(36,40)(37,38);(2,5)(6,8)(9,12,14,27,15,28,13,26)(10,17,29,39,19,35,18,34)(11,21,22,36,24,37,23,25)(16,30,31,40,32,38,20,33);(1,3,4,7)(2,6,5,8)(9,13,15,14)(10,18,19,29)(11,22,24,23)(12,26,28,27)(16,31,32,20)(17,34,35,39)(21,36,37,25)(30,40,38,33)

Polycyclic presentation:

2^9|a^2=fi,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,g^c=gi

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, 7053022);