#7590872

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1256 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fgi,c^2=g,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei

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, 7498068);