#8493264

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

4435 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)(3,4)(5,6)(7,8)(9,10,15,19)(11,16,20,24)(12,14,18,23)(13,22,21,17)(25,26)(27,32)(28,29)(30,33)(31,34)(35,39)(36,37)(38,40);(2,6)(3,8)(11,20)(13,21)(16,24)(17,22)(25,27)(26,32)(28,34)(29,31)(30,37)(33,36)(35,38)(39,40);(9,11,15,20)(10,16,19,24)(12,21,18,13)(14,17,23,22)(25,28)(26,27)(29,34)(30,36)(31,32)(33,35)(37,40)(38,39);(2,6)(4,7)(9,12)(10,14)(11,21)(13,20)(15,18)(16,17)(19,23)(22,24)(25,29,31,27)(26,28,34,32)(30,37,38,35)(33,36,40,39);(1,3)(2,7)(4,6)(5,8)(9,13,15,21)(10,17,19,22)(11,12,20,18)(14,16,23,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40);(1,4,5,7)(2,3,6,8)(9,14)(10,18)(11,17)(12,19)(13,16)(15,23)(20,22)(21,24)(25,28)(26,29)(27,34)(30,36)(31,32)(33,37)(35,40)(38,39)

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=h,e^2=hi,f^2=g,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, 8400460);