#7511154

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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1730 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,d^2=fgi,e^2=i,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

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, 7418350);