#6547482

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1389 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)(3,8)(4,6)(5,7)(10,13)(14,16)(17,18,23,27,22,26,21,28)(19,24,30,41,32,38,31,39)(20,33,34,25,36,42,35,40)(29,43,46,37,45,48,44,47);(1,3,5,4)(2,6,7,8)(9,10,12,13)(11,14,15,16)(17,19,22,32)(18,24,26,38)(20,29,36,45)(21,31,23,30)(25,37,40,47)(27,41,28,39)(33,43,42,48)(34,46,35,44);(10,13)(14,16)(17,20)(18,25)(19,29)(21,34)(22,36)(23,35)(24,37)(26,40)(27,33)(28,42)(30,44)(31,46)(32,45)(38,47)(39,48)(41,43);(1,4,5,3)(2,6,7,8)(11,15)(14,16)(18,26)(24,38)(25,40)(27,28)(33,42)(37,47)(39,41)(43,48);(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,27,26,28)(19,30,32,31)(20,34,36,35)(24,39,38,41)(25,33,40,42)(29,44,45,46)(37,48,47,43)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6454678);