#7720028

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4195 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:

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

Polycyclic presentation:

2^9|a^2=gh,d^2=g,e^2=h,f^2=i,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, 7627224);