#7533151

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	#44929	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1691 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^2=f,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,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, 7440347);