#8093880

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

4602 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,e^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, 8001076);