#8270620

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	#57810	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3924 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=i,d^2=gi,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8177816);