#8186840

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

4168 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)(3,4)(5,9)(6,7)(8,10)(11,12)(13,14)(15,16)(17,18)(19,23)(20,25)(21,29)(22,24)(26,30)(27,28)(31,32)(33,34,38,42)(35,43,45,39)(36,46,44,40)(37,41,47,48);(2,7)(3,10)(9,14)(11,16)(17,19,22,25)(18,23,24,20)(21,27,30,32)(26,31,29,28)(33,35,38,45)(34,39,42,43)(36,37,44,47)(40,41,46,48);(5,13)(9,14)(11,16)(12,15)(18,24)(19,25)(26,29)(27,32)(33,36)(34,40)(35,37)(38,44)(39,41)(42,46)(43,48)(45,47);(1,3)(2,8)(4,7)(5,11)(6,10)(9,15)(12,14)(13,16)(17,20,22,23)(18,25,24,19)(21,28,30,31)(26,32,29,27)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,46)(47,48);(1,4,6,8)(2,3,7,10)(5,12,13,15)(9,11,14,16)(17,21,22,30)(18,26,24,29)(19,27,25,32)(20,28,23,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,46)(47,48);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,31)(28,32)(29,30)(33,37)(34,41)(35,44)(36,45)(38,47)(39,46)(40,43)(42,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=i,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8094036);