#8110603

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

4576 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)(10,12)(13,21)(15,18)(19,23)(25,26)(27,32)(28,37)(29,39)(30,31)(33,38)(34,35)(36,40);(3,7)(6,8)(10,12)(11,17)(16,20)(19,23)(26,31)(32,40)(33,37)(34,39);(1,2)(3,6)(4,5)(7,8)(9,10,14,12)(11,15,17,18)(13,16,21,20)(19,22,23,24)(25,27,30,36)(26,32,31,40)(28,35,38,29)(33,39,37,34);(1,3)(2,6)(4,7)(5,8)(9,11,14,17)(10,15,12,18)(13,19,21,23)(16,22,20,24)(25,28)(26,33)(27,35)(29,36)(30,38)(31,37)(32,39)(34,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,18,17,15)(13,20,21,16)(19,24,23,22)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(9,13,14,21)(10,16,12,20)(11,19,17,23)(15,22,18,24)(25,29,30,35)(26,34,31,39)(27,28,36,38)(32,33,40,37)

Polycyclic presentation:

2^9|c^2=gi,d^2=i,e^2=i,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

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, 8017799);