#8127634

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

4120 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8034830);