#9839263

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

4108 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)(3,5)(4,7)(6,12)(8,10)(9,14)(11,15)(13,16)(17,18)(19,22)(20,27)(21,24)(23,28)(25,31)(26,30)(29,32)(33,34,38,41)(35,39,43,36)(37,45,47,40)(42,44,48,46);(1,3)(2,8)(4,5)(6,11)(7,10)(9,15)(12,13)(14,16)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31)(33,34,38,41)(35,39,43,36)(37,40,47,45)(42,46,48,44);(2,4)(3,10)(9,12)(11,16)(17,18,21,24)(19,22,26,30)(20,23,28,27)(25,29,32,31)(33,35,38,43)(34,36,41,39)(37,42,47,48)(40,44,45,46);(1,4,7,2)(3,5,10,8)(6,12,14,9)(11,13,16,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,45)(47,48);(1,5)(2,3)(4,10)(6,13)(7,8)(9,11)(12,16)(14,15)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,31)(28,32)(33,36,38,39)(34,35,41,43)(37,46,47,44)(40,48,45,42);(1,6,7,14)(2,9,4,12)(3,11,10,16)(5,13,8,15)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,32)(30,31)(33,37)(34,40)(35,42)(36,44)(38,47)(39,46)(41,45)(43,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,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, 9746459);