#8757350

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3979 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=i,d^2=g,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

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, 8664546);