#8096952

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

4561 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,d^2=g,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004148);