#7584597

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1295 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7491793);