#10038493

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

3941 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)(3,8)(4,9)(5,13)(6,7)(10,14)(11,15)(12,16)(19,25)(23,26)(27,31)(30,32)(33,34,39,36)(35,42,37,40)(38,41,46,44)(43,48,45,47);(2,7)(3,10)(4,13)(12,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35,39,37)(34,40,36,42)(38,43,46,45)(41,47,44,48);(2,7)(3,10)(4,13)(12,15)(18,22)(23,26)(24,28)(30,32)(33,34,39,36)(35,40,37,42)(38,41,46,44)(43,47,45,48);(1,3,6,10)(2,8,7,14)(4,11,13,16)(5,12,9,15)(17,19,21,25)(18,23,22,26)(20,27,29,31)(24,30,28,32)(33,36,39,34)(35,40,37,42)(38,44,46,41)(43,47,45,48);(1,4,6,13)(2,9,7,5)(3,11,10,16)(8,15,14,12)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31)(33,37,39,35)(34,40,36,42)(38,43,46,45)(41,48,44,47);(1,5,6,9)(2,4,7,13)(3,12,10,15)(8,11,14,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=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, 9945689);