#8496847

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

4601 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)(3,8)(4,6)(5,7)(9,10,15,14)(11,16,21,20)(12,22,13,17)(18,24,19,23)(28,36)(32,37)(35,40)(38,39);(1,3,6,5)(2,7,4,8)(9,11,15,21)(10,16,14,20)(12,19,13,18)(17,24,22,23)(25,26)(27,31)(28,37)(29,33)(30,34)(32,36)(35,38)(39,40);(9,12,15,13)(10,17,14,22)(11,18,21,19)(16,23,20,24)(25,27)(26,29)(28,35)(30,33)(31,34)(32,38)(36,40)(37,39);(1,2,6,4)(3,7,5,8)(9,13,15,12)(10,17,14,22)(11,19,21,18)(16,23,20,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,40)(37,38);(1,4,6,2)(3,7,5,8)(9,11)(10,16)(12,18)(13,19)(14,20)(15,21)(17,23)(22,24)(25,28)(26,32)(27,35)(29,38)(30,36)(31,39)(33,40)(34,37);(1,5,6,3)(2,7,4,8)(9,14,15,10)(11,20,21,16)(12,22,13,17)(18,24,19,23)(25,29,30,31)(26,33,34,27)(28,38,36,39)(32,40,37,35)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8404043);