#8817760

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	#57199	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4412 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8724956);