#9221191

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

4158 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gi,d^2=gh,e^2=g,f^2=g,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, 9128387);