#8986851

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	#43580	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4007 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)(3,7,11,14)(4,9,13,16)(5,12,15,10)(17,18,22,25)(19,23,26,20)(21,24,29,31)(27,30,32,28)(33,34,40,39)(35,43,37,41)(36,42,46,48)(38,44,45,47);(1,3)(2,7)(4,10)(5,9)(6,11)(8,14)(12,13)(15,16)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32)(33,35,40,37)(34,41,39,43)(36,45,46,38)(42,44,48,47);(2,8)(3,11)(4,13)(5,15)(19,26)(20,23)(21,29)(24,31)(33,36,40,46)(34,42,39,48)(35,45,37,38)(41,44,43,47);(1,3)(2,7)(4,10)(5,9)(6,11)(8,14)(12,13)(15,16)(17,20,22,23)(18,19,25,26)(21,30,29,28)(24,32,31,27)(33,37,40,35)(34,43,39,41)(36,45,46,38)(42,44,48,47);(1,4)(2,9)(3,12)(5,14)(6,13)(7,15)(8,16)(10,11)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,38,40,45)(34,44,39,47)(35,36,37,46)(41,48,43,42);(1,5,6,15)(2,10,8,12)(3,9,11,16)(4,14,13,7)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32)(33,39,40,34)(35,43,37,41)(36,42,46,48)(38,47,45,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

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, 8894047);