#274617

order	512
nilpotency class	3
isoclinism	isoclinic class of #268891
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

643 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)(9,10,14,17,13,16,22,24)(11,18,19,23,15,12,20,21)(25,26,28,32,29,34,30,33)(27,35,36,40,38,31,37,39);(1,3,4,8)(2,5,6,7)(9,11,13,15)(10,12,16,18)(14,19,22,20)(17,21,24,23)(25,27)(26,31)(28,36)(29,38)(30,37)(32,39)(33,40)(34,35);(9,12,13,18)(10,15,16,11)(14,23,22,21)(17,19,24,20)(26,32)(27,36)(28,30)(31,35)(33,34)(37,38);(2,6)(5,7)(10,16)(11,15)(17,24)(19,20)(25,28,29,30)(26,33,34,32)(27,37,38,36)(31,39,35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 181813);