#6612418

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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

1656 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)(19,27,29,43)(20,31,33,24)(21,35,36,25)(23,37,39,30)(28,44,32,42)(34,41,45,47)(38,46,40,48);(9,11)(10,13)(12,16)(14,15)(17,19,21,30,22,29,36,37)(18,23,25,27,26,39,35,43)(20,32,34,48,33,28,45,46)(24,40,41,44,31,38,47,42);(2,5)(6,8)(11,16)(13,14)(17,20)(18,24)(19,28)(21,34)(22,33)(23,38)(25,41)(26,31)(27,42)(29,32)(30,46)(35,47)(36,45)(37,48)(39,40)(43,44);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,16)(13,15)(19,29)(20,33)(23,39)(24,31)(27,43)(30,37)(34,45)(41,47);(11,16)(13,14)(19,29)(23,39)(27,43)(28,32)(30,37)(38,40)(42,44)(46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6519614);