#6458661

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1736 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)(9,10,12,14)(11,13,15,16)(17,18)(19,28)(20,33)(21,25)(22,36)(23,26)(24,31)(27,38)(29,44)(30,43)(32,42)(34,41)(35,48)(37,40)(39,46)(45,47);(9,11)(10,13)(12,15)(14,16)(17,19,23,32)(18,24,26,40)(20,29,21,30)(22,37,38,31)(25,39,33,47)(27,42,36,28)(34,46,35,45)(41,44,48,43);(2,5)(6,8)(11,15)(13,16)(17,20,23,21)(18,25,26,33)(19,29,32,30)(22,34,38,35)(24,39,40,47)(27,41,36,48)(28,43,42,44)(31,45,37,46);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,21,23,20)(18,25,26,33)(19,30,32,29)(22,35,38,34)(24,39,40,47)(27,41,36,48)(28,43,42,44)(31,46,37,45);(18,26)(20,21)(24,40)(27,36)(28,42)(29,30)(34,35)(45,46)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6365857);