#8517412

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	#39430	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4506 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^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, 8424608);