#9778478

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	#41142	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3960 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)(3,7)(4,9)(5,12)(6,11)(8,15)(10,16)(13,14)(33,34)(35,41)(36,44)(37,38)(39,43)(40,45)(42,46)(47,48);(1,3,6,13)(2,7,11,14)(4,10,15,12)(5,9,16,8)(17,18,23,26)(19,20,21,24)(22,25,30,32)(27,28,29,31)(33,34,38,37)(35,39,43,41)(36,44,45,40)(42,47,48,46);(1,4)(2,8)(3,12)(5,14)(6,15)(7,16)(9,11)(10,13)(17,19,23,21)(18,24,26,20)(22,27,30,29)(25,31,32,28)(33,35)(34,39)(36,42)(37,41)(38,43)(40,46)(44,47)(45,48);(1,3)(2,7)(4,12)(5,9)(6,13)(8,16)(10,15)(11,14)(17,20,23,24)(18,19,26,21)(22,28,30,31)(25,27,32,29)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48);(1,4)(2,9)(3,12)(5,7)(6,15)(8,11)(10,13)(14,16)(17,21,23,19)(18,24,26,20)(22,27,30,29)(25,28,32,31)(33,35)(34,41)(36,42)(37,39)(38,43)(40,47)(44,46)(45,48);(1,5)(2,10)(3,9)(4,14)(6,16)(7,15)(8,13)(11,12)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,37,38,34)(35,41,43,39)(36,44,45,40)(42,47,48,46)

Polycyclic presentation:

2^9|b^2=gi,c^2=ghi,d^2=ghi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dghi,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9685674);