#7021379

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

2158 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)(9,11,13,24,15,23,14,25)(10,16,31,36,18,21,19,20)(12,27,28,39,30,22,29,38)(17,33,40,35,34,32,26,37);(3,8)(5,7)(9,12,13,28,15,30,14,29)(10,17,19,26,18,34,31,40)(11,22,24,38,23,27,25,39)(16,32,20,35,21,33,36,37);(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=gi,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,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, 6928575);