#8987043

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	#55054	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4229 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,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, 8894239);