#8137614

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

4603 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8044810);