#8139610

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

4183 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8046806);