#7782710

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	#45417	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4318 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=ghi,d^2=gh,e^2=gh,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7689906);