#6488910

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470402
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1328 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18,23,28)(19,29,33,45)(20,25,21,26)(22,37,30,27)(24,38,40,32)(31,44,35,47)(34,43,36,41)(39,48,42,46);(11,12)(14,15)(17,19,30,38,23,33,22,32)(18,24,37,45,28,40,27,29)(20,31,36,48,21,35,34,46)(25,39,43,47,26,42,41,44);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(17,20,23,21)(18,25,28,26)(19,31,33,35)(22,34,30,36)(24,39,40,42)(27,41,37,43)(29,44,45,47)(32,46,38,48);(9,12,13,11)(10,15,16,14)(17,21,23,20)(18,26,28,25)(19,31,33,35)(22,36,30,34)(24,39,40,42)(27,43,37,41)(29,44,45,47)(32,46,38,48);(1,3,4,7)(2,6,5,8)(10,16)(14,15)(20,21)(25,26)(31,35)(34,36)(39,42)(41,43)(44,47)(46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=hi,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6396106);