#7590713

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1289 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fgi,c^2=g,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7497909);