#7510488

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1808 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)(3,6,4,8)(10,13)(11,15)(17,18,21,24)(19,23,28,31)(20,29,22,25)(26,30,32,27);(1,2)(3,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,26)(20,25)(21,24)(22,29)(23,27)(28,32)(30,31);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,14,13,16)(17,19,21,28)(18,23,24,31)(20,27,22,30)(25,32,29,26);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(18,24)(23,31)(25,29)(26,32);(17,20,21,22)(18,25,24,29)(19,27,28,30)(23,32,31,26)

Polycyclic presentation:

2^9|a^2=fg,c^2=f,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, 7417684);