#7812992

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×#538	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4406 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=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, 7720188);