#8093152

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

4626 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:

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

Polycyclic presentation:

2^9|b^2=g,e^2=gh,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, 8000348);