#7652522

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

6544 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)(5,6,10,8)(7,11,13,14)(9,12,17,16)(15,18,19,20)(21,22)(23,28)(24,26)(25,31)(27,32)(29,34)(30,33)(35,36);(1,2)(3,4)(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,23,27,34)(22,28,32,29)(24,25,33,35)(26,31,30,36);(7,13)(11,14)(15,19)(18,20)(21,23,27,34)(22,29,32,28)(24,25,33,35)(26,36,30,31);(5,8,10,6)(7,14,13,11)(9,12,17,16)(15,18,19,20)(21,24)(22,30)(23,25)(26,32)(27,33)(28,36)(29,31)(34,35);(5,9,10,17)(6,12,8,16)(7,15,13,19)(11,18,14,20)(21,25)(22,31)(23,33)(24,34)(26,29)(27,35)(28,30)(32,36);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,26,27,30)(22,24,32,33)(23,31,34,36)(25,29,35,28)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,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, 7559718);