#9712194

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

4022 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)(3,10)(4,8)(5,13)(6,7)(9,15)(11,12)(14,16)(17,18,22,23)(19,21,25,31)(20,24,29,32)(26,30,27,28)(33,34,39,42)(35,40,38,48)(36,41,46,37)(43,44,45,47);(2,7)(4,10)(9,13)(12,14)(17,19)(18,21)(20,27)(22,25)(23,31)(24,28)(26,29)(30,32)(33,35,39,38)(34,40,42,48)(36,43,46,45)(37,47,41,44);(2,7)(3,8)(9,13)(11,16)(18,23)(19,25)(24,32)(26,27)(33,36,39,46)(34,37,42,41)(35,43,38,45)(40,47,48,44);(1,3,6,8)(2,4,7,10)(5,11,15,16)(9,12,13,14)(17,19)(18,21)(20,26)(22,25)(23,31)(24,30)(27,29)(28,32)(33,36)(34,41)(35,43)(37,42)(38,45)(39,46)(40,44)(47,48);(1,4)(2,8)(3,7)(5,14)(6,10)(9,11)(12,15)(13,16)(17,20,22,29)(18,24,23,32)(19,26,25,27)(21,30,31,28)(33,37,39,41)(34,36,42,46)(35,44,38,47)(40,45,48,43);(1,5,6,15)(2,9,7,13)(3,11,8,16)(4,12,10,14)(17,21)(18,19)(20,28)(22,31)(23,25)(24,27)(26,32)(29,30)(33,38,39,35)(34,40,42,48)(36,45,46,43)(37,47,41,44)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=ghi,d^2=g,e^2=gh,f^2=hi,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, 9619390);