#6476910

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=i,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, 6384106);