#9712323

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

4135 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=ghi,d^2=gi,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 9619519);