#7150026

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994043
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	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1535 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7057222);