#9839284

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
prime radical	#54823	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3963 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9746480);