#7504567

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1690 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)(11,13)(12,14)(15,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,32)(29,30);(1,2)(3,5)(4,6)(7,8)(9,11,12,15)(10,13,14,16)(17,18,20,23)(19,26,27,32)(21,30,29,24)(22,31,25,28);(1,3,4,7)(2,5,6,8)(10,14)(13,16)(17,19)(18,22)(20,27)(21,31)(23,25)(24,32)(26,30)(28,29);(2,6)(5,8)(10,14)(13,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,28)(29,30)(31,32);(18,23)(22,25)(24,30)(26,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,e^d=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, 7411763);