#9850917

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

4127 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)(3,7)(4,5)(6,8)(9,13)(10,11)(12,14)(15,16)(17,18,23,20)(19,26,21,24)(22,28,30,25)(27,31,29,32)(34,39)(40,42)(41,44)(47,48);(1,3,6,12)(2,7,8,14)(4,10,9,15)(5,11,13,16)(17,18,23,20)(19,26,21,24)(22,25,30,28)(27,32,29,31)(33,34)(35,40)(36,44)(37,42)(38,39)(41,45)(43,48)(46,47);(2,8)(4,9)(7,14)(10,15)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,35,38,37)(34,40,39,42)(36,43,45,46)(41,47,44,48);(1,4,6,9)(2,5,8,13)(3,10,12,15)(7,11,14,16)(17,20,23,18)(19,24,21,26)(22,28,30,25)(27,31,29,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,45)(46,48);(1,3)(2,7)(4,10)(5,11)(6,12)(8,14)(9,15)(13,16)(17,21,23,19)(18,24,20,26)(22,27,30,29)(25,32,28,31)(33,36)(34,41)(35,43)(37,46)(38,45)(39,44)(40,47)(42,48);(1,5)(2,9)(3,11)(4,8)(6,13)(7,15)(10,14)(12,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,37,38,35)(34,42,39,40)(36,43,45,46)(41,47,44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=h,e^2=i,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, 9758113);