#6474892

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	#2414	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)(17,18,21,26)(19,25,29,23)(20,31,32,47)(22,35,34,27)(24,38,40,33)(28,42,43,46)(30,37,45,41)(36,44,39,48);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,36)(25,26)(27,37)(31,46)(32,43)(33,44)(34,45)(35,41)(38,48)(39,40)(42,47);(1,3,5,4)(2,7,6,8)(11,15)(13,16)(17,20,22,33,21,32,34,38)(18,24,27,31,26,40,35,47)(19,28,30,44,29,43,45,48)(23,36,37,46,25,39,41,42);(11,15)(13,16)(17,19)(18,25)(20,28)(21,29)(22,30)(23,26)(24,39)(27,41)(31,42)(32,43)(33,44)(34,45)(35,37)(36,40)(38,48)(46,47);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(19,29)(20,32)(23,25)(24,40)(30,45)(31,47)(33,38)(37,41)

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,e^2=fi,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, 6382088);