#8254321

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	#45038	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4354 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)(11,20)(12,16)(13,22)(14,18)(15,19)(17,23)(21,24)(26,30)(27,33)(32,38)(34,35);(1,2,5,4)(3,7,8,6)(11,12)(13,23)(16,20)(17,22)(25,26,29,30)(27,31,33,37)(28,32,36,38)(34,39,35,40);(2,4)(6,7)(9,11,15,12)(10,16,19,20)(13,21,23,18)(14,17,24,22)(25,27)(26,31)(28,35)(29,33)(30,37)(32,40)(34,36)(38,39);(9,12,15,11)(10,16,19,20)(13,18,23,21)(14,17,24,22)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,40)(36,38);(1,3,5,8)(2,6,4,7)(9,13,15,23)(10,17,19,22)(11,21,12,18)(14,16,24,20)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,40)(36,38);(1,4,5,2)(3,7,8,6)(9,14,15,24)(10,18,19,21)(11,22,12,17)(13,16,23,20)(25,28,29,36)(26,32,30,38)(27,34,33,35)(31,39,37,40)

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,d^2=gh,e^2=i,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

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, 8161517);