#7182817

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993899
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1617 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7090013);