#6612171

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

1673 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)(19,28)(20,33)(21,26)(22,37)(23,27)(24,31)(25,35)(29,44)(30,39)(32,43)(34,41)(36,40)(38,47)(42,46)(45,48);(9,11)(10,13)(12,16)(14,15)(17,19,21,30,23,32,22,31)(18,24,37,43,27,39,26,28)(20,29,36,47,35,46,34,45)(25,38,40,44,33,48,41,42);(2,5)(6,8)(11,16)(13,14)(17,20)(18,25)(19,29)(21,36)(22,34)(23,35)(24,38)(26,41)(27,33)(28,42)(30,47)(31,45)(32,46)(37,40)(39,48)(43,44);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,16)(13,15);(11,16)(13,14)(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,34,35,36)(24,28,39,43)(25,40,33,41)(29,45,46,47)(38,44,48,42)

Polycyclic presentation:

2^9|b^2=fhi,d^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dh,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, 6519367);