#8110474

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

4646 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:

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

Polycyclic presentation:

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

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, 8017670);