#8112525

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

4842 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)(13,21)(17,22)(18,23)(19,24)(25,26)(27,33)(28,37)(29,34)(30,40)(31,32)(35,36)(38,39);(1,2,4,6)(3,5,7,8)(9,10)(11,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,24)(26,32)(29,37)(33,40)(36,39);(1,3)(2,5)(4,7)(6,8)(10,15)(11,16)(17,22)(18,23)(25,27,31,30)(26,33,32,40)(28,38,34,35)(29,36,37,39);(1,2)(3,5)(4,6)(7,8)(9,11,14,16)(10,12,15,20)(13,18,21,23)(17,19,22,24)(25,28,31,34)(26,29,32,37)(27,35,30,38)(33,39,40,36);(1,3)(2,5)(4,7)(6,8)(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,29)(26,34)(27,36)(28,32)(30,39)(31,37)(33,35)(38,40);(9,13)(10,17)(11,18)(12,19)(14,21)(15,22)(16,23)(20,24)(25,30,31,27)(26,33,32,40)(28,38,34,35)(29,39,37,36)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=gi,f^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, 8019721);