#6547380

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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

1394 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)(17,18,23,37,22,27,21,26)(19,28,30,40,31,24,32,39)(20,33,34,25,35,41,36,42)(29,44,47,43,46,48,45,38);(1,3,5,4)(2,6,7,8)(9,10,12,13)(11,14,15,16)(17,19,22,31)(18,24,27,28)(20,29,35,46)(21,32,23,30)(25,38,42,43)(26,40,37,39)(33,48,41,44)(34,47,36,45);(10,13)(14,16)(17,20)(18,25)(19,29)(21,34)(22,35)(23,36)(24,38)(26,41)(27,42)(28,43)(30,45)(31,46)(32,47)(33,37)(39,48)(40,44);(1,4,5,3)(2,6,7,8)(11,15)(14,16);(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,26,27,37)(19,30,31,32)(20,34,35,36)(24,39,28,40)(25,41,42,33)(29,45,46,47)(38,48,43,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ei,g^c=gi,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, 6454576);