#9290533

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

4390 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)(3,8)(4,10)(5,6)(7,9)(11,16)(12,13)(14,15)(17,18,22,24)(19,23,27,30)(20,21,28,29)(25,26,31,32)(33,34,39,40)(35,36,37,41)(38,46,47,42)(43,48,45,44);(1,3,7,4)(2,8,9,10)(5,14,11,12)(6,15,16,13)(17,19,22,27)(18,23,24,30)(20,25,28,31)(21,26,29,32)(34,40)(35,37)(42,46)(43,45);(2,9)(3,4)(5,11)(13,15)(18,24)(20,28)(23,30)(25,31)(33,35,39,37)(34,41,40,36)(38,43,47,45)(42,48,46,44);(1,4,7,3)(2,10,9,8)(5,14,11,12)(6,15,16,13)(17,20)(18,21)(19,25)(22,28)(23,26)(24,29)(27,31)(30,32)(33,36,39,41)(34,35,40,37)(38,44,47,48)(42,43,46,45);(1,5,7,11)(2,6,9,16)(3,12,4,14)(8,13,10,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,37,39,35)(34,41,40,36)(38,45,47,43)(42,48,46,44);(1,6)(2,11)(3,13)(4,15)(5,9)(7,16)(8,14)(10,12)(17,21)(18,20)(19,26)(22,29)(23,25)(24,28)(27,32)(30,31)(33,38)(34,42)(35,43)(36,44)(37,45)(39,47)(40,46)(41,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=i,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, 9197729);