#7719054

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4437 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)(9,10,15,17)(11,19,13,16)(12,18,22,14)(20,21,23,24)(25,26,29,33)(27,31,35,30)(28,36,37,32)(34,38,40,39);(1,2)(3,6)(4,7)(5,8)(9,11,15,13)(10,16,17,19)(12,20,22,23)(14,21,18,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,39)(37,40);(10,17)(12,22)(16,19)(20,23)(25,27,29,35)(26,30,33,31)(28,34,37,40)(32,38,36,39);(1,3,5,4)(2,6,8,7)(9,12)(10,18)(11,20)(13,23)(14,17)(15,22)(16,24)(19,21)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,38)(37,40);(9,13,15,11)(10,19,17,16)(12,23,22,20)(14,24,18,21)(25,26)(27,31)(28,32)(29,33)(30,35)(34,39)(36,37)(38,40);(1,4,5,3)(2,7,8,6)(9,14,15,18)(10,12,17,22)(11,21,13,24)(16,20,19,23)(25,28,29,37)(26,32,33,36)(27,34,35,40)(30,38,31,39)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,d^2=gh,e^2=hi,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=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, 7626250);