#6468949

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1784 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)(11,13)(12,14)(15,16)(17,18,22,26,23,27,37,42)(19,28,31,39,32,41,36,24)(20,33,34,48,21,25,35,40)(29,44,45,38,30,43,46,47);(9,11)(10,13)(12,15)(14,16)(17,19,23,32)(18,24,27,39)(20,29,21,30)(22,36,37,31)(25,38,33,47)(26,41,42,28)(34,46,35,45)(40,44,48,43);(2,5)(6,8)(10,14)(11,15)(17,20)(18,25)(19,29)(21,23)(22,34)(24,38)(26,40)(27,33)(28,43)(30,32)(31,45)(35,37)(36,46)(39,47)(41,44)(42,48);(11,15)(13,16)(17,21,23,20)(18,25,27,33)(19,30,32,29)(22,35,37,34)(24,38,39,47)(26,40,42,48)(28,43,41,44)(31,46,36,45);(1,3,4,7)(2,6,5,8)(20,21)(25,33)(29,30)(34,35)(38,47)(40,48)(43,44)(45,46)

Polycyclic presentation:

2^9|a^2=f,b^2=i,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, 6376145);