#8093286

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

4564 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:

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

Polycyclic presentation:

2^9|a^2=gh,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi

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, 8000482);