#9707188

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	#45388	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4159 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=g,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,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, 9614384);