#7589562

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1656 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fgi,d^2=fgi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7496758);