#8691814

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
prime radical	#57149	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4230 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)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(17,18)(19,22)(20,25)(21,30)(23,26)(24,28)(27,29)(31,32)(33,34,37,40)(35,38,41,44)(36,39,43,46)(42,45,47,48);(2,3)(7,9)(8,10)(14,15)(17,19)(18,24)(20,27)(21,31)(22,26)(23,28)(25,32)(29,30)(35,41)(38,44)(42,47)(45,48);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,20,23,21)(18,25,26,30)(19,27,28,31)(22,29,24,32)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,21,23,20)(18,25,26,30)(19,27,28,31)(22,32,24,29)(33,35)(34,38)(36,42)(37,41)(39,45)(40,44)(43,47)(46,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,22,23,24)(18,19,26,28)(20,29,21,32)(25,27,30,31)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,19)(18,22)(20,27)(21,31)(23,28)(24,26)(25,29)(30,32)(33,36,37,43)(34,39,40,46)(35,42,41,47)(38,45,44,48)

Polycyclic presentation:

2^9|a^2=hi,c^2=g,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8599010);