#8475827

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

4248 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)(3,9)(4,6)(5,8)(7,11)(10,16)(12,14)(13,15)(17,18)(19,21)(20,28)(22,23)(24,26)(25,30)(27,31)(29,32)(33,34)(35,43)(36,37)(38,40)(39,42)(41,47)(44,46)(45,48);(2,4)(7,9)(8,12)(15,16)(18,23)(19,26)(25,28)(27,32)(33,35,39,38)(34,40,42,43)(36,44,47,45)(37,48,41,46);(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(33,36)(34,41)(35,44)(37,42)(38,45)(39,47)(40,46)(43,48);(1,3,6,11)(2,7,4,9)(5,13,14,10)(8,16,12,15)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(33,36)(34,41)(35,45)(37,42)(38,44)(39,47)(40,48)(43,46);(1,4,6,2)(3,9,11,7)(5,12,14,8)(10,16,13,15)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,37,39,41)(34,36,42,47)(35,46,38,48)(40,45,43,44);(1,5,6,14)(2,8,4,12)(3,10,11,13)(7,15,9,16)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,27,28,32)(33,38,39,35)(34,40,42,43)(36,45,47,44)(37,48,41,46)

Polycyclic presentation:

2^9|b^2=hi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=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, 8383023);