#8106076

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

4640 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:

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

Polycyclic presentation:

2^9|c^2=gh,d^2=h,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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8013272);