#7710555

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

4310 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)(13,21)(17,22)(18,23)(20,24)(25,26,31,35)(27,36,39,32)(28,33,40,30)(29,37,38,34);(1,2)(3,6)(4,5)(7,8)(9,10,14,15)(11,12,16,19)(13,17,21,22)(18,20,23,24)(25,27)(26,32)(28,37)(29,33)(30,38)(31,39)(34,40)(35,36);(10,15)(12,19)(17,22)(20,24)(25,28,31,40)(26,30,35,33)(27,37,39,34)(29,32,38,36);(9,11,14,16)(10,12,15,19)(13,18,21,23)(17,20,22,24)(25,28)(26,33)(27,37)(29,32)(30,35)(31,40)(34,39)(36,38);(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,22)(21,24)(25,29,31,38)(26,34,35,37)(27,33,39,30)(28,32,40,36);(1,3,4,7)(2,6,5,8)(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,30,31,33)(26,28,35,40)(27,38,39,29)(32,37,36,34)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=i,e^2=h,f^2=g,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, 7617751);