#7658944

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

6860 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:

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

Polycyclic presentation:

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

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, 7566140);