#8505525

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

4614 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)(3,7)(4,6)(5,8)(9,10,15,19)(11,20,12,16)(13,17,23,21)(14,18,24,22)(25,26)(27,35)(28,29)(30,32)(31,34)(33,39)(36,38)(37,40);(1,3,4,8)(2,5,6,7)(11,12)(13,23)(16,20)(17,21)(25,27,31,30)(26,32,34,35)(28,36,39,37)(29,40,33,38);(9,11,15,12)(10,16,19,20)(13,18,23,22)(14,21,24,17)(25,28)(26,33)(27,36)(29,34)(30,37)(31,39)(32,38)(35,40);(2,6)(5,7)(9,12,15,11)(10,16,19,20)(13,22,23,18)(14,21,24,17)(25,28)(26,33)(27,37)(29,34)(30,36)(31,39)(32,40)(35,38);(1,3)(2,5)(4,8)(6,7)(9,13,15,23)(10,17,19,21)(11,18,12,22)(14,20,24,16)(25,29,31,33)(26,28,34,39)(27,38,30,40)(32,37,35,36);(1,2)(3,5)(4,6)(7,8)(9,14)(10,18)(11,21)(12,17)(13,20)(15,24)(16,23)(19,22)(25,30,31,27)(26,32,34,35)(28,37,39,36)(29,40,33,38)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=i,e^2=h,f^2=hi,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, 8412721);