#6469301

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
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	#25723	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1779 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)(19,28)(20,33)(21,25)(22,36)(23,27)(24,31)(26,37)(29,44)(30,42)(32,43)(34,41)(35,48)(38,46)(39,47)(40,45);(9,10)(11,16)(12,14)(13,15)(17,19,22,31,23,32,37,47)(18,24,26,28,27,39,36,43)(20,30,34,46,21,29,35,45)(25,40,41,44,33,38,48,42);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,20,23,21)(18,25,27,33)(19,29,32,30)(22,34,37,35)(24,38,39,40)(26,41,36,48)(28,42,43,44)(31,45,47,46);(10,14)(13,16)(17,21,23,20)(18,25,27,33)(19,30,32,29)(22,35,37,34)(24,38,39,40)(26,41,36,48)(28,42,43,44)(31,46,47,45);(1,4,5,3)(2,6,7,8)(20,21)(25,33)(29,30)(34,35)(38,40)(41,48)(42,44)(45,46)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6376497);