#244255

order	512
nilpotency class	3
isoclinism	isoclinic class of #233367
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

714 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151451);