#6611501

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1557 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fgh,c^2=gi,d^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, 6518697);