#9230879

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

4178 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)(5,15)(7,10)(8,13)(11,14)(12,16)(17,18)(19,20)(21,29)(22,25)(23,27)(24,32)(26,30)(28,31)(33,34,39,43)(35,44,46,40)(36,47,38,41)(37,48,45,42);(1,3)(2,8)(4,11)(5,12)(6,10)(7,13)(9,16)(14,15)(17,19,22,27)(18,20,25,23)(21,26,32,31)(24,28,29,30)(33,35,39,46)(34,40,43,44)(36,45,38,37)(41,42,47,48);(1,4,7,5)(2,9,10,15)(3,11,13,12)(6,14,8,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,29)(31,32)(33,36,39,38)(34,41,43,47)(35,45,46,37)(40,42,44,48);(1,5,7,4)(2,9,10,15)(3,12,13,11)(6,14,8,16)(17,20,22,23)(18,19,25,27)(21,30,32,28)(24,31,29,26)(33,37)(34,42)(35,36)(38,46)(39,45)(40,47)(41,44)(43,48);(1,2)(3,8)(4,9)(5,15)(6,13)(7,10)(11,16)(12,14)(17,19)(18,20)(21,31)(22,27)(23,25)(24,30)(26,32)(28,29)(33,38,39,36)(34,41,43,47)(35,45,46,37)(40,48,44,42);(1,6,7,8)(2,3,10,13)(4,14,5,16)(9,11,15,12)(17,21,22,32)(18,24,25,29)(19,26,27,31)(20,28,23,30)(33,34)(35,40)(36,41)(37,48)(38,47)(39,43)(42,45)(44,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=gh,e^2=hi,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, 9138075);