#9718506

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	#46103	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3945 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)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,22,23)(19,21,25,31)(20,24,29,32)(26,30,27,28)(33,34)(35,45)(36,41)(37,42)(38,43)(39,44)(40,47)(46,48);(3,9)(7,13)(10,15)(14,16)(17,19)(18,21)(20,27)(22,25)(23,31)(24,28)(26,29)(30,32)(33,35)(34,40)(36,38)(37,46)(39,47)(41,48)(42,43)(44,45);(2,6)(7,13)(8,11)(14,16)(18,23)(19,25)(24,32)(26,27)(33,36,39,37)(34,41,44,42)(35,38,47,46)(40,48,45,43);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,18)(19,21)(20,24)(22,23)(25,31)(26,30)(27,28)(29,32)(33,37,39,36)(34,42,44,41)(35,46,47,38)(40,43,45,48);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,20,22,29)(18,24,23,32)(19,26,25,27)(21,30,31,28)(33,38,39,46)(34,43,44,48)(35,36,47,37)(40,42,45,41);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,21)(18,19)(20,28)(22,31)(23,25)(24,27)(26,32)(29,30)(33,34)(35,40)(36,41)(37,42)(38,48)(39,44)(43,46)(45,47)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=g,e^2=gi,f^2=hi,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, 9625702);