#8181146

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

4578 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:

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

Polycyclic presentation:

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

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, 8088342);