#257967

order	512
nilpotency class	3
isoclinism	isoclinic class of #247367
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ 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)(3,8)(4,6)(5,7)(9,10,13,17)(11,19,12,15)(14,23,22,18)(16,20,24,21)(25,26)(27,31)(28,33)(29,32)(30,34)(35,38)(36,40)(37,39);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,13)(14,20)(16,23)(17,19)(18,24)(21,22)(25,27)(26,31)(28,36)(29,35)(30,37)(32,38)(33,40)(34,39);(1,4,5,3)(2,6,7,8)(9,12,13,11)(10,16,17,24)(14,20,22,21)(15,18,19,23)(26,32)(27,35)(28,29)(33,34)(36,37)(38,40);(2,7)(6,8)(10,17)(11,12)(18,23)(20,21)(25,28,30,29)(26,33,34,32)(27,35,37,36)(31,38,39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 165163);