#8132823

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

4410 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)(11,17)(12,20)(13,18)(14,15)(16,21)(19,22)(23,24)(25,26)(27,33)(28,31)(29,34)(30,32)(35,40)(36,38)(37,39);(1,2)(3,6)(4,5)(7,8)(10,15)(11,18)(16,20)(19,24)(25,27,31,30)(26,32,28,33)(29,35,38,37)(34,39,36,40);(2,5)(6,8)(10,15)(11,18)(12,21)(22,23)(25,27,31,30)(26,33,28,32)(29,37,38,35)(34,39,36,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,19)(14,18)(15,17)(16,22)(20,23)(21,24)(25,28,31,26)(27,33,30,32)(29,36,38,34)(35,40,37,39);(1,2)(3,6)(4,5)(7,8)(9,12,14,21)(10,16,15,20)(11,19,18,24)(13,22,17,23)(25,29,31,38)(26,34,28,36)(27,35,30,37)(32,39,33,40);(1,3,4,7)(2,6,5,8)(9,13)(10,11)(12,22)(14,17)(15,18)(16,19)(20,24)(21,23)(25,30,31,27)(26,33,28,32)(29,37,38,35)(34,40,36,39)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=gh,e^2=g,f^2=ghi,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, 8040019);