#8533100

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

4721 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=hi,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440296);