#6476908

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1765 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6384104);