#6474885

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1384 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)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,31)(21,26)(22,37)(23,27)(25,35)(28,33)(29,43)(30,42)(32,45)(34,46)(36,39)(38,44)(40,47)(41,48);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,32)(21,23)(22,30)(25,40)(26,27)(28,39)(29,34)(31,45)(33,36)(35,47)(37,42)(38,41)(43,46)(44,48);(1,3,5,4)(2,7,6,8)(11,15)(13,16)(17,20,33,48,23,34,22,35)(18,25,37,46,27,41,28,31)(19,29,36,47,21,32,30,44)(24,38,42,45,26,40,39,43);(11,15)(13,16)(17,21,23,19)(18,26,27,24)(20,32,34,29)(22,36,33,30)(25,40,41,38)(28,42,37,39)(31,45,46,43)(35,47,48,44);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(18,27)(19,21)(20,34)(28,37)(30,36)(35,48)(38,40)(43,45)

Polycyclic presentation:

2^9|c^2=fgi,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=ei,e^d=ei

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, 6382081);