#6612334

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574270
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

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

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, 6519530);