#8183749

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	#35133	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4072 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8090945);