#8254682

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	#56390	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4216 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)(12,20)(13,14)(15,19)(16,22)(18,23)(21,24)(25,26,31,34)(27,29,33,30)(28,36,39,32)(35,38,40,37);(1,2,5,4)(3,7,8,6)(9,11)(10,15)(12,21)(13,17)(14,19)(16,24)(18,22)(20,23)(25,27,31,33)(26,30,34,29)(28,35,39,40)(32,38,36,37);(2,4)(6,7)(9,10,14,13)(11,15,19,17)(12,16,22,20)(18,23,21,24)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,38)(39,40);(9,12)(10,16)(11,18)(13,20)(14,22)(15,23)(17,24)(19,21)(25,28,31,39)(26,32,34,36)(27,35,33,40)(29,37,30,38);(1,3,5,8)(2,6,4,7)(9,13,14,10)(11,15,19,17)(12,20,22,16)(18,23,21,24)(25,29,31,30)(26,33,34,27)(28,37,39,38)(32,40,36,35);(1,4,5,2)(3,7,8,6)(9,11)(10,17)(12,18)(13,15)(14,19)(16,24)(20,23)(21,22)(25,30,31,29)(26,27,34,33)(28,38,39,37)(32,35,36,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=gh,e^2=hi,f^2=i,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, 8161878);