#10181033

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

4055 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=hi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10088229);