#6733265

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1506 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=fg,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,e^d=ei,h^e=hi

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, 6640461);