#7644338

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

6848 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)(7,13)(8,16)(9,10)(11,14)(12,17)(15,18)(19,20)(21,22)(23,31)(24,26)(25,28)(27,30)(29,36)(32,33)(34,35);(1,2,3,4)(7,14)(11,13)(15,20)(18,19)(21,23,27,25)(22,28,30,31)(24,34,29,32)(26,33,36,35);(6,10)(11,13)(12,16)(18,19)(23,25)(28,31)(32,34)(33,35);(5,7,9,14)(6,11,10,13)(8,15,17,20)(12,18,16,19)(21,24,27,29)(22,26,30,36)(23,32,25,34)(28,35,31,33);(5,6)(7,11)(8,12)(9,10)(13,14)(15,18)(16,17)(19,20)(21,25,27,23)(22,28,30,31)(24,34,29,32)(26,35,36,33);(5,8,9,17)(6,12,10,16)(7,15,14,20)(11,18,13,19)(21,26)(22,29)(23,33)(24,30)(25,35)(27,36)(28,32)(31,34)

Polycyclic presentation:

2^9|b^2=g,d^2=hi,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, 7551534);