#6996803

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996799
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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2280 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=f,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,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, 6903999);