#8528140

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4318 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)(11,19,20,16)(12,22,23,17)(13,18,24,21)(25,26)(27,35)(28,29)(30,32)(31,34)(33,40)(36,37)(38,39);(2,3)(6,7)(10,15)(11,20)(17,22)(18,21)(25,27,31,30)(26,32,34,35)(28,38,33,36)(29,37,40,39);(1,2,5,3)(4,6,8,7)(10,15)(12,23)(16,19)(18,21)(25,28,31,33)(26,29,34,40)(27,36,30,38)(32,39,35,37);(1,3,5,2)(4,7,8,6)(9,11,14,20)(10,16,15,19)(12,21,23,18)(13,22,24,17)(25,29)(26,33)(27,37)(28,34)(30,39)(31,40)(32,36)(35,38);(1,4,5,8)(2,6,3,7)(9,12,14,23)(10,17,15,22)(11,21,20,18)(13,19,24,16)(25,30,31,27)(26,32,34,35)(28,38,33,36)(29,39,40,37);(9,13)(10,18)(11,17)(12,19)(14,24)(15,21)(16,23)(20,22)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=i,d^2=h,e^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, 8435336);