#8734804

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

4304 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8642000);