#6817248

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814815
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

1929 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fgh,d^2=f,e^2=fh,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di,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, 6724444);