#6625005

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1704 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,18,23,28)(19,29,31,40)(20,34,37,41)(21,26,22,27)(24,33,39,32)(25,35,42,36)(30,44,45,38)(43,47,48,46);(2,6)(7,8)(10,14)(11,12)(17,19,23,31)(18,24,28,39)(20,30,37,45)(21,32,22,33)(25,38,42,44)(26,40,27,29)(34,48,41,43)(35,46,36,47);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,38)(26,34)(27,41)(28,42)(29,43)(31,45)(32,46)(33,47)(39,44)(40,48);(1,4,5,3)(2,8,6,7)(11,12)(15,16)(19,31)(24,39)(29,40)(30,45)(32,33)(38,44)(43,48)(46,47);(9,12,13,11)(10,16,14,15)(17,21,23,22)(18,26,28,27)(19,32,31,33)(20,35,37,36)(24,40,39,29)(25,34,42,41)(30,46,45,47)(38,48,44,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=fgh,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,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, 6532201);