#6594146

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1412 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)(20,34)(21,25)(22,27)(23,33)(24,35)(26,29)(30,44)(31,38)(32,41)(36,39)(37,46)(40,43)(42,47)(45,48);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,29,31,22,32,21,33)(18,23,25,41,27,38,26,28)(20,30,43,45,36,46,35,47)(24,37,39,48,40,44,34,42);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,20,21,35,22,36,29,43)(18,24,26,34,27,40,25,39)(19,30,33,47,32,46,31,45)(23,37,28,42,38,44,41,48);(9,11)(10,13)(12,15)(14,16)(17,21,22,29)(18,25,27,26)(19,31,32,33)(20,35,36,43)(23,28,38,41)(24,39,40,34)(30,45,46,47)(37,42,44,48);(11,15)(13,16)(19,32)(23,38)(28,41)(30,46)(31,33)(37,44)(42,48)(45,47)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6501342);