#8254553

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	#45373	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4204 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)(11,17)(12,14)(13,16)(15,19)(18,24)(20,22)(21,23)(26,30)(28,35)(31,36)(34,39);(1,2,5,4)(3,7,8,6)(10,12)(15,17)(16,20)(23,24)(27,33)(31,36)(34,39)(38,40);(2,4)(6,7)(9,11,14,19)(10,15,12,17)(13,21,22,18)(16,24,20,23)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(9,10,14,12)(11,15,19,17)(13,16,22,20)(18,23,21,24)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(1,3,5,8)(2,6,4,7)(9,12,14,10)(11,17,19,15)(13,20,22,16)(18,24,21,23)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40);(1,4,5,2)(3,7,8,6)(9,13,14,22)(10,16,12,20)(11,18,19,21)(15,23,17,24)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=h,e^2=hi,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, 8161749);