#7470929

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309000
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1218 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,26)(20,24)(21,30)(22,25)(23,28)(27,32)(29,31)(33,34,36,41,37,40,44,39)(35,38,42,47,43,46,48,45);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,22,28)(18,23,25,26)(20,27,21,29)(24,31,30,32)(33,35,37,43)(34,38,40,46)(36,42,44,48)(39,45,41,47);(17,19)(18,23)(20,29)(21,27)(22,28)(24,32)(25,26)(30,31)(34,39)(36,44)(38,45)(40,41)(42,48)(46,47);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20,22,21)(18,24,25,30)(19,27,28,29)(23,31,26,32)(34,40)(38,46)(39,41)(45,47);(2,8)(6,9)(7,16)(14,15)(17,21,22,20)(18,24,25,30)(19,27,28,29)(23,32,26,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=fi,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7378125);