#8291663

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

4072 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=i,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^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8198859);