#8256594

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	#45922	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3967 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=g,d^2=gh,e^2=ghi,f^2=hi,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, 8163790);