#7718444

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

4173 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=h,e^2=i,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, 7625640);