#9723167

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4665 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)(3,7)(4,6)(5,8)(9,10,15,17)(11,16,14,24)(12,18,21,13)(19,20,22,23)(25,26)(27,37)(28,31)(29,34)(30,35)(32,36)(33,39)(38,40);(1,2,4,6)(3,5,8,7)(9,11,15,14)(10,16,17,24)(12,19,21,22)(13,23,18,20)(25,27,32,39)(26,33,36,37)(28,38,29,30)(31,35,34,40);(1,2)(3,5)(4,6)(7,8)(10,17)(12,21)(16,24)(19,22)(25,28,32,29)(26,31,36,34)(27,38,39,30)(33,35,37,40);(1,3,4,8)(2,5,6,7)(9,12)(10,18)(11,19)(13,17)(14,22)(15,21)(16,20)(23,24)(25,29,32,28)(26,34,36,31)(27,30,39,38)(33,40,37,35);(9,13,15,18)(10,12,17,21)(11,20,14,23)(16,22,24,19)(25,30)(26,35)(27,28)(29,39)(31,37)(32,38)(33,34)(36,40);(9,14,15,11)(10,16,17,24)(12,22,21,19)(13,23,18,20)(25,31,32,34)(26,28,36,29)(27,35,39,40)(30,33,38,37)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=g,e^2=h,f^2=ghi,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.

Relationships

maximal normal subgroup of	#72219279, #72219281, #72219285
maximal quotient of	#72219273

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9630363);