#8110592

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4220 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017788);