#9863316

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

4230 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^2=hi,e^2=hi,f^2=gh,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, 9770512);