#8528124

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
prime radical	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4246 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)(3,5)(4,7)(6,8)(9,10,14,15)(11,18,16,12)(13,17,22,23)(19,24,21,20)(27,33)(30,37)(34,39)(38,40);(2,3)(6,7)(10,15)(11,16)(17,23)(19,21)(25,26,29,32)(27,30,33,37)(28,35,36,31)(34,40,39,38);(1,2,5,3)(4,6,8,7)(10,15)(12,18)(17,23)(20,24)(25,27)(26,30)(28,34)(29,33)(31,38)(32,37)(35,40)(36,39);(1,3,5,2)(4,7,8,6)(9,11,14,16)(10,12,15,18)(13,21,22,19)(17,24,23,20)(25,28,29,36)(26,31,32,35)(27,34,33,39)(30,38,37,40);(1,4)(2,6)(3,7)(5,8)(9,12)(10,16)(11,15)(13,20)(14,18)(17,21)(19,23)(22,24)(25,27)(26,30)(28,34)(29,33)(31,38)(32,37)(35,40)(36,39);(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,21)(18,24)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=i,d^2=h,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, 8435320);