#8374798

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

4117 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8281994);