#8274152

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

4194 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8181348);