#8463672

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

4024 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8370868);