#7513480

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1399 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)(3,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18,21,22)(19,23,26,30)(20,27,29,24)(25,28,32,31);(1,3)(2,4)(5,7)(6,8)(10,13)(11,15)(17,19,21,26)(18,23,22,30)(20,28,29,31)(24,25,27,32);(10,13)(14,16)(17,20,21,29)(18,24,22,27)(19,25,26,32)(23,31,30,28);(1,4,6,5)(2,3,7,8)(9,11,12,15)(10,14,13,16);(1,5,6,4)(2,3,7,8)(17,18,21,22)(19,23,26,30)(20,24,29,27)(25,31,32,28)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=f,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,h^b=hi,h^c=hi

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, 7420676);