#7498562

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309020
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1370 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=hi,c^2=i,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7405758);