#9668342

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

4207 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)(3,5)(4,7)(6,12)(8,10)(9,14)(11,15)(13,16)(17,18)(19,24)(20,26)(21,27)(22,32)(23,25)(28,29)(30,31)(33,34,38,42)(35,36,40,39)(37,45,47,41)(43,48,46,44);(1,3)(2,8)(4,5)(6,11)(7,10)(9,15)(12,13)(14,16)(17,19,23,22)(18,24,25,32)(20,28,27,30)(21,31,26,29)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,48)(46,47);(2,4)(3,10)(9,12)(11,16)(18,25)(20,27)(24,32)(28,30)(33,35,38,40)(34,39,42,36)(37,43,47,46)(41,48,45,44);(1,4,7,2)(3,5,10,8)(6,12,14,9)(11,13,16,15)(17,20,23,27)(18,26,25,21)(19,28,22,30)(24,29,32,31)(33,35)(34,36)(37,46)(38,40)(39,42)(41,48)(43,47)(44,45);(1,5)(2,3)(4,10)(6,13)(7,8)(9,11)(12,16)(14,15)(17,21,23,26)(18,27,25,20)(19,29,22,31)(24,28,32,30)(33,36)(34,40)(35,42)(37,44)(38,39)(41,46)(43,45)(47,48);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,22,23,19)(18,24,25,32)(20,30,27,28)(21,31,26,29)(33,37,38,47)(34,41,42,45)(35,43,40,46)(36,44,39,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gi,d^2=ghi,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,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9575538);