#8821930

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

4058 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)(3,9,10,7)(4,8,12,13)(11,15,16,14)(33,34,39,44)(35,42,37,40)(36,46,45,41)(38,43,47,48);(2,5)(7,9)(8,13)(14,15)(17,18)(19,22)(20,28)(21,24)(23,29)(25,27)(26,31)(30,32)(33,35,39,37)(34,40,44,42)(36,47,45,38)(41,48,46,43);(3,10)(4,12)(7,9)(8,13)(17,18,21,24)(19,25,27,22)(20,23,29,28)(26,31,32,30)(33,36)(34,41)(35,38)(37,47)(39,45)(40,43)(42,48)(44,46);(1,3,6,10)(2,7,5,9)(4,11,12,16)(8,14,13,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32)(33,37,39,35)(34,42,44,40)(36,47,45,38)(41,48,46,43);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31)(33,38,39,47)(34,43,44,48)(35,45,37,36)(40,46,42,41);(1,5,6,2)(3,9,10,7)(4,13,12,8)(11,15,16,14)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,46)(47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=g,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,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, 8729126);