#6567421

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1295 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)(9,10,14,13)(11,16,12,15)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,33)(25,38)(30,39)(31,41)(32,40)(35,42)(36,44)(37,43)(45,46)(47,48);(1,3,5,4)(2,6,8,7)(9,11,14,12)(10,15,13,16)(17,19,22,32,23,31,21,33)(18,24,26,41,28,40,27,29)(20,35,38,47,36,30,37,46)(25,42,34,45,43,39,44,48);(1,3,5,4)(2,7,8,6)(9,10,14,13)(11,16,12,15)(17,20,21,37,23,36,22,38)(18,25,27,44,28,43,26,34)(19,30,33,47,31,35,32,46)(24,39,29,45,40,42,41,48);(1,3,5,4)(2,7,8,6)(9,12,14,11)(10,15,13,16)(19,31)(20,36)(24,40)(25,43)(29,41)(32,33)(34,44)(37,38);(1,4,5,3)(2,6,8,7)(9,13,14,10)(11,16,12,15)(17,21,23,22)(18,26,28,27)(19,32,31,33)(20,37,36,38)(24,29,40,41)(25,34,43,44)(30,46,35,47)(39,45,42,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=f,d^2=fg,e^2=fgi,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=ei,d^c=di,e^d=ehi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6474617);