#6612627

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1660 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18,23,27)(19,28,30,40)(20,32,33,25)(21,26,22,37)(24,35,39,31)(29,44,47,43)(34,42,36,41)(38,46,48,45);(9,11)(10,13)(12,16)(14,15)(17,19,21,35,23,30,22,31)(18,24,37,40,27,39,26,28)(20,29,36,45,33,47,34,46)(25,38,41,44,32,48,42,43);(2,5)(6,8)(11,16)(13,14)(17,20)(18,25)(19,29)(21,36)(22,34)(23,33)(24,38)(26,42)(27,32)(28,43)(30,47)(31,46)(35,45)(37,41)(39,48)(40,44);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,16)(13,15)(19,30)(20,33)(24,39)(25,32)(28,40)(31,35)(34,36)(41,42);(11,16)(13,14)(17,21,23,22)(18,26,27,37)(19,31,30,35)(20,34,33,36)(24,40,39,28)(25,41,32,42)(29,45,47,46)(38,43,48,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,d^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ehi,d^c=dhi,e^c=ei,e^d=eh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6519823);