#7644363

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6708 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,d^b=dhi,e^c=ehi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7551559);