#8662911

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

4209 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)(3,10)(4,12)(5,7)(6,9)(8,13)(11,15)(14,16)(17,18,23,26)(19,22,24,20)(21,25,30,32)(27,31,29,28)(34,39)(35,41)(36,43)(44,45);(1,3,6,5)(2,7,9,10)(4,11,13,14)(8,15,12,16)(17,19)(18,22)(20,26)(21,29)(23,24)(25,28)(27,30)(31,32)(33,34,38,39)(35,37,41,46)(36,40,43,47)(42,45,48,44);(17,20,23,22)(18,24,26,19)(21,28,30,31)(25,29,32,27)(33,35,38,41)(34,37,39,46)(36,42,43,48)(40,45,47,44);(1,4)(2,8)(3,11)(5,14)(6,13)(7,15)(9,12)(10,16)(17,21)(18,25)(19,27)(20,28)(22,31)(23,30)(24,29)(26,32)(33,34)(35,37)(36,40)(38,39)(41,46)(42,45)(43,47)(44,48);(1,2)(3,7)(4,8)(5,10)(6,9)(11,15)(12,13)(14,16)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,30)(31,32)(33,36)(34,40)(35,42)(37,45)(38,43)(39,47)(41,48)(44,46);(1,5,6,3)(2,7,9,10)(4,11,13,14)(8,16,12,15)(17,22,23,20)(18,24,26,19)(21,28,30,31)(25,27,32,29)(33,37)(34,41)(35,39)(36,44)(38,46)(40,42)(43,45)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8570107);