#8112504

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

4622 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)(5,8)(9,10,14,12)(11,17,18,15)(13,20,21,16)(19,22,24,23)(29,37)(33,38)(34,39)(35,40);(1,2,4,6)(3,5,7,8)(10,12)(15,17)(16,20)(22,23)(25,26)(27,28)(29,33)(30,31)(32,36)(34,35)(37,38)(39,40);(1,3)(2,5)(4,7)(6,8)(11,18)(15,17)(19,24)(22,23)(26,31)(27,32)(33,38)(34,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,17)(13,19)(14,18)(16,22)(20,23)(21,24)(25,27,30,32)(26,28,31,36)(29,34,37,39)(33,35,38,40);(1,3)(2,5)(4,7)(6,8)(9,12,14,10)(11,17,18,15)(13,20,21,16)(19,23,24,22)(25,28)(26,32)(27,31)(29,35)(30,36)(33,39)(34,38)(37,40);(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,29)(26,33)(27,34)(28,35)(30,37)(31,38)(32,39)(36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=g,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, 8019700);