#8963660

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4198 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8870856);