#9712233

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	#42138	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=ghi,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,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, 9619429);