#9724835

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

4249 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)(3,11)(4,13)(5,10)(6,14)(7,9)(8,12)(15,16)(17,18)(19,24)(20,26)(21,27)(22,32)(23,25)(28,29)(30,31)(33,34)(35,36)(37,45)(38,42)(39,40)(41,47)(43,48)(44,46);(1,3,7,12)(2,8,9,11)(4,5,14,15)(6,10,13,16)(17,19,23,22)(18,24,25,32)(20,28,27,30)(21,31,26,29)(33,35,38,40)(34,39,42,36)(37,43,47,46)(41,48,45,44);(2,9)(3,12)(4,14)(10,16)(18,25)(20,27)(24,32)(28,30)(33,36,38,39)(34,40,42,35)(37,44,47,48)(41,46,45,43);(1,4)(2,6)(3,5)(7,14)(8,10)(9,13)(11,16)(12,15)(17,20,23,27)(18,26,25,21)(19,28,22,30)(24,29,32,31)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,46)(47,48);(1,5)(2,10)(3,4)(6,8)(7,15)(9,16)(11,13)(12,14)(17,21,23,26)(18,27,25,20)(19,29,22,31)(24,28,32,30)(33,35)(34,36)(37,46)(38,40)(39,42)(41,48)(43,47)(44,45);(1,6)(2,4)(3,10)(5,8)(7,13)(9,14)(11,15)(12,16)(17,22,23,19)(18,24,25,32)(20,30,27,28)(21,31,26,29)(33,37)(34,41)(35,43)(36,44)(38,47)(39,48)(40,46)(42,45)

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9632031);