#9706284

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	#46669	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4014 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)(3,6)(4,9)(5,10)(7,8)(11,15)(12,14)(13,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32)(33,34,39,37)(35,43,45,40)(36,46,47,41)(38,48,44,42);(1,3)(2,6)(4,5)(7,11)(8,15)(9,10)(12,13)(14,16)(18,22)(23,25)(24,28)(30,31)(33,35,39,45)(34,40,37,43)(36,44,47,38)(41,42,46,48);(2,8)(3,11)(4,12)(10,16)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31)(33,36,39,47)(34,41,37,46)(35,38,45,44)(40,48,43,42);(1,4,7,12)(2,9,8,14)(3,5,11,13)(6,10,15,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,36)(34,41)(35,44)(37,46)(38,45)(39,47)(40,42)(43,48);(1,5)(2,10)(3,12)(4,11)(6,14)(7,13)(8,16)(9,15)(17,18)(19,23)(20,28)(21,22)(24,29)(25,27)(26,31)(30,32)(33,37,39,34)(35,40,45,43)(36,46,47,41)(38,48,44,42);(1,6,7,15)(2,11,8,3)(4,10,12,16)(5,9,13,14)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)(33,38,39,44)(34,42,37,48)(35,36,45,47)(40,46,43,41)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=g,d^2=gh,e^2=ghi,f^2=h,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, 9613480);