#8630818

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

4183 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)(3,10)(4,12)(5,7)(6,8)(9,14)(11,15)(13,16)(17,18)(19,24)(20,25)(21,23)(22,26)(27,31)(28,32)(29,30)(33,34,40,38)(35,39,43,41)(36,42,37,47)(44,46,45,48);(1,3,7,6)(2,8,5,10)(4,13,14,11)(9,16,12,15)(17,19,23,27)(18,24,21,31)(20,28,30,26)(22,25,32,29)(33,35,40,43)(34,41,38,39)(36,44,37,45)(42,48,47,46);(2,5)(8,10)(9,12)(15,16)(18,21)(19,27)(25,29)(26,28)(33,36,40,37)(34,42,38,47)(35,44,43,45)(39,46,41,48);(1,4)(2,9)(3,11)(5,12)(6,13)(7,14)(8,15)(10,16)(17,20,23,30)(18,25,21,29)(19,26,27,28)(22,31,32,24)(33,37,40,36)(34,42,38,47)(35,45,43,44)(39,46,41,48);(1,5,7,2)(3,8,6,10)(4,12,14,9)(11,15,13,16)(17,21,23,18)(19,24,27,31)(20,29,30,25)(22,28,32,26)(33,38,40,34)(35,41,43,39)(36,42,37,47)(44,46,45,48);(1,6,7,3)(2,8,5,10)(4,11,14,13)(9,16,12,15)(17,22)(18,26)(19,25)(20,24)(21,28)(23,32)(27,29)(30,31)(33,39,40,41)(34,43,38,35)(36,46,37,48)(42,45,47,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=ghi,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8538014);