#10451638

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

3965 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10358834);