#9719133

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#37824	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4273 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,5,3)(4,7,8,6)(9,10)(11,16)(12,13)(14,22)(15,17)(18,23)(19,21)(20,24)(25,26)(27,34)(28,30)(29,33)(31,35)(32,39)(36,40)(37,38);(2,3)(6,7)(9,11,15,20)(10,16,17,24)(12,21,23,14)(13,19,18,22)(25,27)(26,32)(28,36)(29,37)(30,38)(31,39)(33,40)(34,35);(1,3,5,2)(4,7,8,6)(10,17)(11,20)(12,23)(19,22)(25,28,31,29)(26,30,35,33)(27,36,39,37)(32,38,34,40);(1,4,5,8)(2,6,3,7)(9,12)(10,18)(11,21)(13,17)(14,20)(15,23)(16,22)(19,24)(25,29,31,28)(26,33,35,30)(27,37,39,36)(32,40,34,38);(9,13,15,18)(10,12,17,23)(11,22,20,19)(14,24,21,16)(25,27)(26,34)(28,36)(29,37)(30,40)(31,39)(32,35)(33,38);(9,14,15,21)(10,19,17,22)(11,12,20,23)(13,16,18,24)(25,30,31,33)(26,28,35,29)(27,38,39,40)(32,36,34,37)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9626329);