#7720204

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4188 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)(9,10,16,15)(11,12,18,14)(13,17,23,22)(19,20,24,21)(28,36)(32,37)(35,39)(38,40);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,15)(13,19)(16,18)(17,21)(20,22)(23,24)(26,30)(27,33)(32,37)(35,39);(9,12,16,14)(10,11,15,18)(13,21,23,20)(17,24,22,19)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(9,13,16,23)(10,17,15,22)(11,19,18,24)(12,20,14,21);(1,4,5,2)(3,7,8,6)(9,14,16,12)(10,18,15,11)(13,21,23,20)(17,24,22,19)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,39)(37,40);(9,15,16,10)(11,12,18,14)(13,22,23,17)(19,20,24,21)(25,28,29,36)(26,32,30,37)(27,35,33,39)(31,38,34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=g,f^2=hi,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, 7627400);