#8115318

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

4408 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022514);