#8271982

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

4146 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:

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

Polycyclic presentation:

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

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, 8179178);