#9230864

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	#48715	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4114 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,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, 9138060);