#9726497

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

4083 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)(3,12)(4,9)(5,8)(6,10)(7,11)(13,14)(15,16)(17,18)(19,24)(20,23)(21,25)(22,26)(27,30)(28,31)(29,32)(33,34)(35,36)(37,45)(38,39)(40,43)(41,47)(42,46)(44,48);(1,3,7,5)(2,8,11,12)(4,13,10,15)(6,14,9,16)(17,19,23,26)(18,24,20,22)(21,29,31,27)(25,32,28,30)(34,39)(36,40)(41,45)(44,46);(18,20)(19,26)(25,28)(27,29)(33,35,38,43)(34,40,39,36)(37,42,47,48)(41,46,45,44);(1,4,7,10)(2,9,11,6)(3,13,5,15)(8,16,12,14)(17,20,23,18)(19,22,26,24)(21,28,31,25)(27,32,29,30)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,46)(47,48);(1,5,7,3)(2,8,11,12)(4,15,10,13)(6,14,9,16)(17,21,23,31)(18,25,20,28)(19,27,26,29)(22,32,24,30)(33,36)(34,35)(37,46)(38,40)(39,43)(41,48)(42,45)(44,47);(1,6,7,9)(2,10,11,4)(3,14,5,16)(8,15,12,13)(17,22)(18,19)(20,26)(21,30)(23,24)(25,29)(27,28)(31,32)(33,37,38,47)(34,41,39,45)(35,42,43,48)(36,44,40,46)

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,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, 9633693);