#6458391

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1714 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

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, 6365587);