#8093775

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4281 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)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(26,30)(31,36)(32,37)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(26,30)(27,33)(32,37)(34,39);(2,5)(6,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,22)(13,16)(14,17)(18,21)(19,24)(20,23)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)

Polycyclic presentation:

2^9|c^2=gh,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8000971);