#8840319

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	#50739	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4487 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8747515);