#8630810

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

4026 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)(33,34,39,40)(35,37,36,41)(38,42,46,47)(43,45,44,48);(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)(34,40)(35,36)(42,47)(43,44);(2,5)(8,10)(9,12)(15,16)(18,21)(19,27)(25,29)(26,28)(33,35,39,36)(34,41,40,37)(38,43,46,44)(42,48,47,45);(1,4,7,14)(2,9,5,12)(3,11,6,13)(8,15,10,16)(17,20,23,30)(18,25,21,29)(19,26,27,28)(22,31,32,24)(33,36,39,35)(34,37,40,41)(38,44,46,43)(42,45,47,48);(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)(33,37)(34,35)(36,40)(38,45)(39,41)(42,43)(44,47)(46,48);(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)(33,38,39,46)(34,42,40,47)(35,43,36,44)(37,45,41,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=ghi,d^2=h,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, 8538006);