#242830

order	512
nilpotency class	3
isoclinism	isoclinic class of #233220
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

682 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)(9,10,14,22,13,18,12,17)(11,15,19,16,21,24,20,23)(25,26,28,32,30,34,29,33)(27,31,35,40,37,38,36,39);(2,5)(6,8)(9,11)(10,15)(12,20)(13,21)(14,19)(16,22)(17,23)(18,24)(25,27,30,37)(26,31,34,38)(28,35,29,36)(32,40,33,39);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,19)(13,21)(14,20)(15,22)(17,24)(18,23)(26,32)(27,35)(28,29)(31,38)(33,34)(36,37);(9,12,13,14)(10,17,18,22)(11,19,21,20)(15,16,24,23)(25,28,30,29)(26,32,34,33)(27,36,37,35)(31,39,38,40)

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=e,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,c^b=cg,d^b=dh,g^b=gi,d^c=di,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, 150026);