#7021377

order	512
nilpotency class	3
isoclinism	isoclinic class of #7021375
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

2217 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)(9,10,13,19,15,18,14,31)(11,20,24,21,23,36,25,16)(12,17,28,26,30,34,29,40)(22,35,38,33,27,37,39,32);(1,2)(3,7)(4,6)(5,8)(9,10)(11,21)(12,26)(13,31)(14,19)(15,18)(16,23)(17,28)(20,24)(22,37)(25,36)(27,35)(29,34)(30,40)(32,39)(33,38);(1,3)(2,5)(4,8)(6,7)(9,11,14,25,15,23,13,24)(10,16,19,20,18,21,31,36)(12,27,29,38,30,22,28,39)(17,33,26,37,34,32,40,35);(3,8)(5,7)(9,12,14,29,15,30,13,28)(10,17,31,40,18,34,19,26)(11,22,25,39,23,27,24,38)(16,32,36,37,21,33,20,35);(2,6)(5,7)(10,18)(11,23)(17,34)(19,31)(22,27)(24,25)(26,40)(38,39)

Polycyclic presentation:

2^9|a^2=f,c^2=g,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,e^c=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, 6928573);