#8818978

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4235 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gi,d^2=i,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, 8726174);