#8371310

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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=g,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=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, 8278506);