#6996809

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

2200 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=fi,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, 6904005);