#8509001

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

4436 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gi,d^2=hi,e^2=i,f^2=gi,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, 8416197);