#6469253

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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

1511 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)(3,8)(4,6)(5,7)(11,15)(13,16)(17,18,22,27,23,28,37,44)(19,29,32,40,33,43,36,24)(20,25,34,41,21,26,35,42)(30,45,47,39,31,46,48,38);(9,10)(11,16)(12,14)(13,15)(17,19,23,33)(18,24,28,40)(20,31,21,30)(22,36,37,32)(25,39,26,38)(27,43,44,29)(34,47,35,48)(41,45,42,46);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,20)(18,25)(19,30)(21,23)(22,34)(24,38)(26,28)(27,41)(29,45)(31,33)(32,47)(35,37)(36,48)(39,40)(42,44)(43,46);(10,14)(13,16)(17,21,23,20)(18,26,28,25)(19,31,33,30)(22,35,37,34)(24,39,40,38)(27,42,44,41)(29,46,43,45)(32,48,36,47);(1,4,5,3)(2,6,7,8)(20,21)(25,26)(30,31)(34,35)(38,39)(41,42)(45,46)(47,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=i,e^2=ghi,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,d^c=di,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, 6376449);