#8726221

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3931 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=gh,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8633417);