#8818528

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	#57191	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4266 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8725724);