#7511145

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×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1790 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:

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

Polycyclic presentation:

2^9|a^2=fgi,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,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, 7418341);