#8505615

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

4661 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)(3,8)(4,7)(5,6)(9,10,15,12)(11,16,21,19)(13,22,24,17)(14,18,20,23)(25,26,31,30)(27,34,29,32)(28,33,36,38)(35,40,37,39);(2,6)(3,7)(9,11)(10,16)(12,19)(13,14)(15,21)(17,23)(18,22)(20,24)(28,36)(33,38)(35,37)(39,40);(9,10,15,12)(11,19,21,16)(13,17,24,22)(14,18,20,23)(25,27,31,29)(26,32,30,34)(28,35,36,37)(33,39,38,40);(2,6)(4,8)(9,12,15,10)(11,19,21,16)(13,22,24,17)(14,18,20,23)(25,28)(26,33)(27,35)(29,37)(30,38)(31,36)(32,39)(34,40);(1,3,5,7)(2,4,6,8)(9,13)(10,17)(11,20)(12,22)(14,21)(15,24)(16,18)(19,23)(25,29,31,27)(26,34,30,32)(28,37,36,35)(33,40,38,39);(1,4)(2,7)(3,6)(5,8)(9,14,15,20)(10,18,12,23)(11,13,21,24)(16,22,19,17)(25,30,31,26)(27,34,29,32)(28,38,36,33)(35,40,37,39)

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=i,e^2=gh,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, 8412811);