#9712190

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	#38676	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4017 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)(3,8)(4,11)(5,13)(6,7)(9,15)(10,12)(14,16)(17,18,23,24)(19,21,25,29)(20,22,28,31)(26,27,30,32)(33,34,39,37)(35,41,36,40)(38,42,46,45)(43,48,44,47);(1,3,6,11)(2,4,7,8)(5,10,15,14)(9,12,13,16)(17,19,23,25)(18,21,24,29)(20,26,28,30)(22,27,31,32)(33,35,39,36)(34,40,37,41)(38,43,46,44)(42,47,45,48);(2,7)(4,8)(9,13)(12,16)(18,24)(19,25)(20,28)(27,32)(33,34,39,37)(35,40,36,41)(38,42,46,45)(43,47,44,48);(1,4,6,8)(2,3,7,11)(5,12,15,16)(9,10,13,14)(17,20)(18,22)(19,26)(21,27)(23,28)(24,31)(25,30)(29,32)(33,36,39,35)(34,41,37,40)(38,44,46,43)(42,48,45,47);(1,3)(2,8)(4,7)(5,14)(6,11)(9,12)(10,15)(13,16)(17,21,23,29)(18,25,24,19)(20,27,28,32)(22,30,31,26)(33,37,39,34)(35,41,36,40)(38,42,46,45)(43,47,44,48);(1,5)(2,9)(3,10)(4,12)(6,15)(7,13)(8,16)(11,14)(17,22)(18,20)(19,27)(21,26)(23,31)(24,28)(25,32)(29,30)(33,38,39,46)(34,42,37,45)(35,43,36,44)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=ghi,d^2=g,e^2=gh,f^2=ghi,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, 9619386);