#7646721

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

6558 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

(1,2,3,4)(5,6,10,14)(7,13,9,11)(8,12,18,16)(15,20,17,19)(22,27)(23,31)(29,30)(32,33);(1,2)(3,4)(21,22,26,27)(23,28,31,34)(24,29,25,30)(32,35,33,36);(5,7,10,9)(6,11,14,13)(8,15,18,17)(12,19,16,20)(21,22)(23,28)(24,29)(25,30)(26,27)(31,34)(32,35)(33,36);(5,6)(7,11)(8,16)(9,13)(10,14)(12,18)(15,20)(17,19)(21,23)(22,28)(24,32)(25,33)(26,31)(27,34)(29,35)(30,36);(5,8)(6,12)(7,15)(9,17)(10,18)(11,19)(13,20)(14,16)(21,24,26,25)(22,29,27,30)(23,32,31,33)(28,35,34,36);(5,9,10,7)(6,13,14,11)(8,17,18,15)(12,20,16,19)(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,c^2=hi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

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, 7553917);