#6518977

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500186
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1456 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)(4,6)(5,8)(17,18,23,28,22,24,21,27)(19,29,31,38,32,42,37,25)(20,33,34,41,35,26,36,40)(30,44,47,39,46,43,45,48);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(18,24)(25,38)(26,33)(27,28)(29,42)(39,48)(40,41)(43,44);(10,13)(11,15)(17,19,22,32)(18,25,24,38)(20,30,35,46)(21,31,23,37)(26,39,33,48)(27,29,28,42)(34,45,36,47)(40,44,41,43);(1,3,5,4)(2,7,8,6)(9,11,12,15)(10,14,13,16)(17,20)(18,26)(19,30)(21,36)(22,35)(23,34)(24,33)(25,39)(27,41)(28,40)(29,43)(31,47)(32,46)(37,45)(38,48)(42,44);(1,4,5,3)(2,6,8,7)(10,13)(14,16)(17,21,22,23)(18,27,24,28)(19,31,32,37)(20,34,35,36)(25,29,38,42)(26,40,33,41)(30,45,46,47)(39,44,48,43)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^2=fhi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6426173);