#8107491

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

4492 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

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, 8014687);