#242021

order	512
nilpotency class	3
isoclinism	isoclinic class of #234099
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

765 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)(3,6)(4,5)(7,8)(9,10)(11,20)(12,16)(13,18)(14,19)(15,23)(17,22)(21,24)(25,26,29,33)(27,30,35,38)(28,31,37,32)(34,39,36,40);(2,5)(6,8)(9,11,14,12)(10,16,19,20)(13,22,15,21)(17,18,24,23)(25,27,29,35)(26,30,33,38)(28,34,37,36)(31,39,32,40);(9,12,14,11)(10,17,19,24)(13,22,15,21)(16,18,20,23)(26,31)(27,34)(28,37)(32,33)(35,36)(39,40);(1,3,4,7)(2,6,5,8)(9,13,14,15)(10,18,19,23)(11,21,12,22)(16,17,20,24)(27,35)(30,38)(34,36)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 149217);