#9782826

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

4128 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=h,d^2=g,e^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, 9690022);