#8095058

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

4415 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002254);