#9874938

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

4193 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)(3,9)(4,5)(6,7)(8,12)(10,15)(11,16)(13,14)(17,18,23,25)(19,27,21,24)(20,30,26,22)(28,29,31,32)(34,38)(36,44)(39,41)(42,48);(2,7)(4,13)(8,9)(10,16)(17,19,23,21)(18,24,25,27)(20,28,26,31)(22,29,30,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(18,25)(20,26)(24,27)(28,31)(33,34,37,38)(35,39,43,41)(36,40,44,45)(42,46,48,47);(1,3,6,12)(2,8,7,9)(4,10,13,16)(5,11,14,15)(17,20,23,26)(18,22,25,30)(19,28,21,31)(24,29,27,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,4,6,13)(2,5,7,14)(3,10,12,16)(8,11,9,15)(17,21,23,19)(18,24,25,27)(20,31,26,28)(22,29,30,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,5)(2,4)(3,11)(6,14)(7,13)(8,10)(9,16)(12,15)(17,22)(18,26)(19,29)(20,25)(21,32)(23,30)(24,31)(27,28)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=i,d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9782134);