#316620

order	512
nilpotency class	3
isoclinism	isoclinic class of #262883
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

639 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=egi,c^2=hi,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cg,d^b=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 223816);