#256523

order	512
nilpotency class	3
isoclinism	isoclinic class of #246751
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

721 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)(11,19)(12,17)(13,16)(14,18)(15,21)(20,24)(22,23)(25,26)(27,33)(28,39)(29,36)(30,35)(31,34)(32,38)(37,40);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,21)(13,22)(14,20)(16,24)(17,19)(18,23)(25,27)(26,32)(28,31)(29,37)(30,38)(33,35)(34,40)(36,39);(1,4,5,3)(2,6,7,8)(10,16)(11,20)(13,14)(17,18)(21,22)(23,24)(25,28)(26,33)(27,29)(30,37)(31,38)(32,35)(34,39)(36,40);(2,7)(6,8)(10,17)(15,19)(16,18)(23,24)(25,29,30,31)(26,34,35,36)(27,37,38,28)(32,40,33,39)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 163719);