#7504849

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500695
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1460 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7412045);