#9382280

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

4196 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9289476);