#6537837

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

1523 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,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, 6445033);