#8841743

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

4320 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:

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

Polycyclic presentation:

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

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, 8748939);