#8425468

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

4205 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=ghi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8332664);