#9726412

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

4123 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)(3,5)(4,11)(6,14)(7,8)(9,16)(10,15)(12,13)(17,18,22,26)(19,24,29,23)(20,21,25,31)(27,30,32,28)(33,34,38,42)(35,41,45,39)(36,37,46,40)(43,47,48,44);(1,3,7,4)(2,5,8,11)(6,10,16,12)(9,13,14,15)(17,19)(18,23)(20,27)(21,28)(22,29)(24,26)(25,32)(30,31)(33,35,38,45)(34,39,42,41)(36,43,46,48)(37,44,40,47);(2,8)(3,4)(9,14)(10,12)(33,36,38,46)(34,40,42,37)(35,43,45,48)(39,47,41,44);(1,4,7,3)(2,5,8,11)(6,12,16,10)(9,13,14,15)(17,20,22,25)(18,21,26,31)(19,27,29,32)(23,28,24,30)(33,36)(34,37)(35,43)(38,46)(39,44)(40,42)(41,47)(45,48);(1,5)(2,3)(4,8)(6,15)(7,11)(9,12)(10,14)(13,16)(17,19)(18,24)(20,27)(21,30)(22,29)(23,26)(25,32)(28,31)(33,35)(34,41)(36,43)(37,47)(38,45)(39,42)(40,44)(46,48);(1,6,7,16)(2,9,8,14)(3,10,4,12)(5,13,11,15)(17,21)(18,25)(19,28)(20,26)(22,31)(23,32)(24,27)(29,30)(33,37)(34,36)(35,44)(38,40)(39,43)(41,48)(42,46)(45,47)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=gh,f^2=i,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, 9633608);