#6657499

order	512
nilpotency class	3
isoclinism	isoclinic class of #6636940
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1490 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fgi,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6564695);