#8720826

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

4100 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)(3,9)(4,10)(5,8)(6,7)(11,14)(12,13)(15,16)(17,18)(19,24)(20,28)(21,25)(22,27)(23,29)(26,30)(31,32)(33,34,40,43)(35,44,39,41)(36,38,37,42)(45,47,46,48);(2,7)(3,10)(8,16)(11,13)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,35,40,39)(34,41,43,44)(36,46,37,45)(38,47,42,48);(2,7)(3,10)(8,16)(11,13)(20,29)(23,28)(26,32)(30,31)(33,36,40,37)(34,42,43,38)(35,45,39,46)(41,47,44,48);(1,3)(2,4)(5,13)(6,10)(7,9)(8,14)(11,15)(12,16)(17,20,21,29)(18,23,25,28)(19,26,27,32)(22,30,24,31)(33,37,40,36)(34,38,43,42)(35,46,39,45)(41,48,44,47);(1,4)(2,3)(5,14)(6,9)(7,10)(8,13)(11,16)(12,15)(17,19)(18,24)(20,26)(21,27)(22,25)(23,31)(28,30)(29,32)(33,38)(34,36)(35,47)(37,43)(39,48)(40,42)(41,46)(44,45);(1,5,6,15)(2,8,7,16)(3,11,10,13)(4,12,9,14)(17,18)(19,22)(20,28)(21,25)(23,29)(24,27)(26,31)(30,32)(33,39,40,35)(34,41,43,44)(36,46,37,45)(38,48,42,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8628022);