#6605687

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1776 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)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,23,28,21,26)(19,29,31,39,32,42,37,24)(20,33,34,41,36,25,35,40)(30,44,46,38,47,43,45,48);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,30)(21,31)(22,37)(23,32)(25,38)(26,29)(27,42)(28,39)(33,48)(34,45)(35,46)(36,47)(40,44)(41,43);(2,6)(5,8)(10,14)(11,15)(17,20)(18,25)(19,30)(21,35)(22,34)(23,36)(24,38)(26,41)(27,40)(28,33)(29,43)(31,46)(32,47)(37,45)(39,48)(42,44);(9,11)(10,13)(12,15)(14,16)(17,21,23,22)(18,26,28,27)(19,31,32,37)(20,34,36,35)(24,29,39,42)(25,40,33,41)(30,45,47,46)(38,44,48,43);(11,15)(13,16)(17,21,23,22)(18,26,28,27)(19,31,32,37)(20,35,36,34)(24,29,39,42)(25,41,33,40)(30,46,47,45)(38,43,48,44)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6512883);