#8824369

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

4073 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=hi,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8731565);