#9779352

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	#45468	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4224 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9686548);