#8358681

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4349 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,d^2=gi,e^2=gh,f^2=g,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, 8265877);