#7728225

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

4364 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7635421);