#7709492

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

4444 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)(3,5,7,8)(9,10,14,15)(11,13,17,22)(12,19,20,16)(18,24,23,21)(25,26,31,28)(27,35,36,32)(29,33,39,37)(30,38,40,34);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,16)(14,15)(17,22)(18,21)(19,20)(23,24)(25,27,31,36)(26,32,28,35)(29,34,39,38)(30,33,40,37);(10,15)(11,17)(12,20)(21,24)(27,36)(29,39)(32,35)(33,37);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,18)(14,17)(15,22)(16,21)(19,24)(20,23)(25,28,31,26)(27,35,36,32)(29,37,39,33)(30,38,40,34);(9,12,14,20)(10,16,15,19)(11,18,17,23)(13,21,22,24)(25,29)(26,33)(27,34)(28,37)(30,35)(31,39)(32,40)(36,38);(9,13)(10,11)(12,21)(14,22)(15,17)(16,18)(19,23)(20,24)(25,30)(26,34)(27,33)(28,38)(29,35)(31,40)(32,39)(36,37)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=gh,e^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7616688);