#9290537

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

4180 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18,21,22)(19,23,27,25)(20,28,29,24)(26,31,32,30)(33,34)(35,40)(36,42)(37,38)(39,41)(43,48)(44,45)(46,47);(2,6)(4,13)(7,9)(11,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,39,36)(34,40,41,42)(37,46,43,44)(38,47,48,45);(2,6)(3,10)(8,12)(11,16)(18,22)(23,25)(24,28)(30,31)(34,41)(35,36)(37,43)(45,47);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36,39,35)(34,42,41,40)(37,46,43,44)(38,47,48,45);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31)(33,37,39,43)(34,38,41,48)(35,44,36,46)(40,45,42,47);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,38)(34,43)(35,45)(36,47)(37,41)(39,48)(40,46)(42,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=gi,e^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9197733);