#6605565

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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

1940 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)(17,18,23,28)(19,29,32,42)(20,33,34,25)(21,27,22,26)(24,37,39,31)(30,44,45,43)(35,40,36,41)(38,46,47,48);(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)(33,47)(34,45)(35,46)(36,48)(40,43)(41,44);(2,6)(5,8)(10,14)(11,15)(17,20)(18,25)(19,30)(21,35)(22,36)(23,34)(24,38)(26,40)(27,41)(28,33)(29,43)(31,46)(32,45)(37,48)(39,47)(42,44);(9,11)(10,13)(12,15)(14,16)(20,34)(25,33)(30,45)(35,36)(38,47)(40,41)(43,44)(46,48);(11,15)(13,16)(17,21,23,22)(18,26,28,27)(19,31,32,37)(20,35,34,36)(24,29,39,42)(25,40,33,41)(30,46,45,48)(38,43,47,44)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6512761);