#7659395

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

6700 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:

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

Polycyclic presentation:

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

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, 7566591);