#6464411

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1524 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)(17,18,23,28)(19,29,33,44)(20,25,21,26)(22,36,38,27)(24,37,41,32)(30,45,31,46)(34,43,35,42)(39,48,40,47);(9,10)(11,15)(12,13)(14,16)(17,19,23,33)(18,24,28,41)(20,31,21,30)(22,37,38,32)(25,40,26,39)(27,44,36,29)(34,47,35,48)(42,45,43,46);(2,5)(6,8)(10,13)(14,15)(17,20,23,21)(18,25,28,26)(19,30,33,31)(22,34,38,35)(24,39,41,40)(27,42,36,43)(29,45,44,46)(32,47,37,48);(9,11,12,16)(10,14,13,15)(17,21,23,20)(18,26,28,25)(19,31,33,30)(22,35,38,34)(24,40,41,39)(27,43,36,42)(29,46,44,45)(32,48,37,47);(1,3,4,7)(2,6,5,8)(20,21)(25,26)(30,31)(34,35)(39,40)(42,43)(45,46)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6371607);