#8126094

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

4722 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8033290);