#9712243

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3810 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=ghi,d^2=gh,e^2=g,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, 9619439);