#9851008

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

3968 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=gh,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, 9758204);