#8461169

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

4114 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8368365);