#8552362

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4114 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:

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

Polycyclic presentation:

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

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, 8459558);