#9203830

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

4236 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9111026);