#9685559

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	#50951	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(18,24)(20,25)(26,31)(28,32)(33,34,40,44)(35,39,37,41)(36,46,48,42)(38,43,47,45);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,18)(19,20)(21,25)(22,26)(23,24)(27,28)(29,32)(30,31)(33,35,40,37)(34,41,44,39)(36,45,48,43)(38,46,47,42);(2,3)(7,9)(8,10)(14,15)(17,19,23,21)(18,20,24,25)(22,27,30,29)(26,28,31,32)(33,36,40,48)(34,42,44,46)(35,45,37,43)(38,39,47,41);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,23,25)(18,19,24,21)(22,28,30,32)(26,27,31,29)(33,37,40,35)(34,39,44,41)(36,45,48,43)(38,46,47,42);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,21,23,19)(18,25,24,20)(22,27,30,29)(26,28,31,32)(33,38)(34,43)(35,42)(36,39)(37,46)(40,47)(41,48)(44,45);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,39)(34,35)(36,47)(37,44)(38,48)(40,41)(42,45)(43,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,d^2=g,e^2=gh,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, 9592755);