#8943832

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

4270 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=i,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=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, 8851028);