#8132900

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

4191 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8040096);