#6553864

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1352 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)(17,18,22,27)(19,28,31,40)(20,32,35,44)(21,25,34,26)(23,38,41,30)(24,36,43,37)(29,42,33,39)(45,46,48,47);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,30,22,31,34,38)(18,23,26,28,27,41,25,40)(20,33,37,46,35,29,36,47)(24,42,32,48,43,39,44,45);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,20,34,36,22,35,21,37)(18,24,25,44,27,43,26,32)(19,29,38,46,31,33,30,47)(23,39,40,48,41,42,28,45);(1,4,5,3)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(20,35)(24,43)(29,33)(32,44)(36,37)(39,42)(45,48)(46,47);(11,15)(14,16)(17,21,22,34)(18,25,27,26)(19,30,31,38)(20,36,35,37)(23,40,41,28)(24,32,43,44)(29,46,33,47)(39,45,42,48)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fi,d^2=fgi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^c=di,e^c=ei,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6461060);