#8211638

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	#38087	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4367 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 8118834);