#8429263

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	#42565	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4372 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)(15,21)(16,19)(22,23)(25,26)(27,32)(28,29)(30,34)(31,35)(33,38)(36,39)(37,40);(1,2)(3,8)(4,6)(5,7)(9,10)(11,15)(12,16)(13,14)(17,22)(18,21)(19,20)(23,24)(27,35)(31,32)(36,40)(37,39);(1,3,5,4)(2,6,7,8)(25,27)(26,31)(28,36)(29,37)(30,35)(32,34)(33,39)(38,40);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,19)(13,14)(16,20)(17,23)(18,21)(22,24)(25,27)(26,32)(28,36)(29,39)(30,35)(31,34)(33,37)(38,40);(9,11,13,18)(10,15,14,21)(12,17,20,24)(16,22,19,23)(25,28)(26,33)(27,36)(29,34)(30,38)(31,39)(32,37)(35,40);(1,4,5,3)(2,6,7,8)(9,12,13,20)(10,16,14,19)(11,17,18,24)(15,22,21,23)(25,29,30,33)(26,28,34,38)(27,37,35,39)(31,36,32,40)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8336459);