#9206638

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	#41627	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3958 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=gh,e^2=hi,f^2=h,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, 9113834);