#9712281

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

4098 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=ghi,d^2=gi,e^2=gh,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, 9619477);