#6615760

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1442 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)(3,7)(4,8)(5,6)(17,18,23,37,22,26,21,27)(19,28,30,24,32,40,31,39)(20,33,35,42,34,25,36,41)(29,44,47,48,45,43,46,38);(2,6)(7,8)(9,10,12,13)(11,14,15,16)(17,19,22,32)(18,24,26,39)(20,29,34,45)(21,30,23,31)(25,38,33,48)(27,40,37,28)(35,46,36,47)(41,43,42,44);(2,6)(7,8)(10,13)(11,15)(17,20)(18,25)(19,29)(21,36)(22,34)(23,35)(24,38)(26,33)(27,42)(28,43)(30,47)(31,46)(32,45)(37,41)(39,48)(40,44);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(18,26)(20,34)(24,39)(27,37)(28,40)(29,45)(35,36)(46,47);(1,4,5,3)(2,8,6,7)(10,13)(11,15)(17,21,22,23)(18,27,26,37)(19,30,32,31)(20,35,34,36)(24,40,39,28)(25,41,33,42)(29,46,45,47)(38,43,48,44)

Polycyclic presentation:

2^9|a^2=fg,b^2=hi,d^2=gi,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,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, 6522956);