#6816974

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

2050 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=gh,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, 6724170);