#9706336

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
prime radical	#53684	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3961 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,d^2=gh,e^2=hi,f^2=gh,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, 9613532);