#8115765

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	#38388	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4587 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:

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

Polycyclic presentation:

2^9|c^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^c=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, 8022961);