#9718193

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	#2555	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3962 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)(3,9)(4,14)(5,8)(6,10)(7,13)(11,15)(12,16)(17,18)(19,29)(20,21)(22,26)(23,31)(24,28)(25,32)(27,30)(34,40)(35,38)(36,42)(45,47);(1,3,8,13)(2,9,5,7)(4,12,6,11)(10,15,14,16)(17,19)(18,25)(20,30)(21,31)(22,23)(24,32)(26,27)(28,29)(33,34)(35,43)(36,37)(38,41)(39,40)(42,48)(44,45)(46,47);(1,4,8,6)(2,10,5,14)(3,11,13,12)(7,15,9,16)(17,20,24,22)(18,26,28,21)(19,30,32,23)(25,27,29,31)(33,35,39,38)(34,41,40,43)(36,46,42,44)(37,45,48,47);(1,5,8,2)(3,7,13,9)(4,10,6,14)(11,16,12,15)(17,21,24,26)(18,20,28,22)(19,31,32,27)(23,25,30,29)(33,36,39,42)(34,37,40,48)(35,44,38,46)(41,47,43,45);(1,6,8,4)(2,10,5,14)(3,12,13,11)(7,15,9,16)(17,22,24,20)(18,26,28,21)(19,30,32,23)(25,31,29,27)(33,37)(34,42)(35,45)(36,40)(38,47)(39,48)(41,44)(43,46);(1,7)(2,3)(4,15)(5,13)(6,16)(8,9)(10,11)(12,14)(17,23,24,30)(18,27,28,31)(19,22,32,20)(21,25,26,29)(33,38,39,35)(34,41,40,43)(36,46,42,44)(37,47,48,45)

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,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, 9625389);