#7513869

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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1365 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)(3,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18,21,23)(19,24,28,30)(20,25,22,26)(27,32,29,31);(1,3,6,2)(4,7,5,8)(18,23)(19,27)(20,22)(24,31)(28,29)(30,32);(10,13)(14,16)(17,19,21,28)(18,24,23,30)(20,27,22,29)(25,32,26,31);(1,4,6,5)(2,7,3,8)(10,13)(14,16)(17,20,21,22)(18,25,23,26)(19,27,28,29)(24,32,30,31);(1,5,6,4)(2,7,3,8)(9,11,12,15)(10,14,13,16)(18,23)(24,30)(25,26)(31,32)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=f,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,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, 7421065);