#8749982

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	#36696	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3963 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=hi,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8657178);