#9839121

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

4247 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,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, 9746317);