#8505297

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

4147 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)(17,18,23,20)(19,24,29,27)(21,30,32,25)(22,26,28,31)(33,34,39,37)(35,42,40,36)(38,41,47,45)(43,46,48,44);(1,2,5,6)(3,7,9,13)(4,8,12,14)(10,15,11,16)(17,19)(18,24)(20,27)(21,22)(23,29)(25,31)(26,30)(28,32)(34,37)(36,42)(41,45)(44,46);(17,18,23,20)(19,27,29,24)(21,25,32,30)(22,26,28,31)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,46)(47,48);(1,3)(2,7)(4,11)(5,9)(6,13)(8,16)(10,12)(14,15)(17,20,23,18)(19,27,29,24)(21,30,32,25)(22,26,28,31)(33,36,39,42)(34,40,37,35)(38,46,47,44)(41,43,45,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,15)(11,16)(12,14)(17,21)(18,25)(19,28)(20,30)(22,29)(23,32)(24,26)(27,31)(33,37,39,34)(35,42,40,36)(38,45,47,41)(43,46,48,44);(1,4,5,12)(2,8,6,14)(3,10,9,11)(7,15,13,16)(17,22,23,28)(18,26,20,31)(19,21,29,32)(24,30,27,25)(33,38,39,47)(34,41,37,45)(35,43,40,48)(36,44,42,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,d^2=hi,e^2=h,f^2=gi,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, 8412493);