#8726197

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	#40125	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4087 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)(19,27)(20,22)(21,24)(23,25)(26,31)(28,29)(30,32)(33,34,39,43)(35,38,42,40)(36,41,46,37)(44,45,48,47);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,19,23,21)(18,24,25,27)(20,30,31,28)(22,29,26,32)(33,35,39,42)(34,40,43,38)(36,44,46,48)(37,45,41,47);(2,6)(7,14)(8,11)(15,16)(18,25)(20,31)(24,27)(28,30)(33,36,39,46)(34,41,43,37)(35,44,42,48)(38,45,40,47);(1,2)(3,7)(4,11)(5,6)(8,13)(9,16)(10,14)(12,15)(17,20,23,31)(18,26,25,22)(19,28,21,30)(24,29,27,32)(33,36)(34,37)(35,44)(38,47)(39,46)(40,45)(41,43)(42,48);(1,3,5,10)(2,7,6,14)(4,12,13,9)(8,16,11,15)(17,21,23,19)(18,27,25,24)(20,30,31,28)(22,29,26,32)(33,37)(34,36)(35,45)(38,48)(39,41)(40,44)(42,47)(43,46);(1,4,5,13)(2,8,6,11)(3,9,10,12)(7,15,14,16)(17,22,23,26)(18,20,25,31)(19,29,21,32)(24,30,27,28)(33,38,39,40)(34,42,43,35)(36,45,46,47)(37,44,41,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=h,e^2=hi,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, 8633393);