#7018528

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996315
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	#59017	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1590 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)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,25,23,26,21,30)(19,27,29,31,28,32,20,24);(1,3,4,7)(2,5,6,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,28)(18,24,26,31)(20,21,29,22)(25,32,30,27);(2,6)(5,8)(9,11)(10,13)(12,16)(14,15)(19,28)(20,29)(24,31)(27,32);(11,16)(13,15)(17,20,22,19,23,29,21,28)(18,24,25,27,26,31,30,32);(10,14)(13,15)(17,21,23,22)(18,25,26,30)(19,20,28,29)(24,27,31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6925724);