#8093487

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

3987 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)(3,8)(4,9)(5,12)(6,7)(10,15)(11,14)(13,16)(17,18,24,28)(19,27,23,25)(20,21,22,26)(29,32,31,30)(33,34,38,42)(35,43,46,39)(36,40,37,41)(44,48,45,47);(3,11)(4,13)(8,14)(9,16)(17,19,24,23)(18,25,28,27)(20,31,22,29)(21,32,26,30)(33,35,38,46)(34,39,42,43)(36,44,37,45)(40,47,41,48);(2,7)(3,11)(9,16)(10,12)(17,20,24,22)(18,26,28,21)(19,29,23,31)(25,32,27,30)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,3)(2,8)(4,12)(5,9)(6,11)(7,14)(10,13)(15,16)(17,21,24,26)(18,20,28,22)(19,30,23,32)(25,29,27,31)(33,35)(34,39)(36,44)(37,45)(38,46)(40,47)(41,48)(42,43);(1,4,6,13)(2,9,7,16)(3,12,11,10)(5,14,15,8)(17,22,24,20)(18,26,28,21)(19,31,23,29)(25,32,27,30)(33,36,38,37)(34,40,42,41)(35,44,46,45)(39,47,43,48);(1,5)(2,10)(3,9)(4,14)(6,15)(7,12)(8,13)(11,16)(17,23,24,19)(18,27,28,25)(20,31,22,29)(21,32,26,30)(33,37,38,36)(34,41,42,40)(35,45,46,44)(39,48,43,47)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000683);