#256719

order	512
nilpotency class	3
isoclinism	isoclinic class of #246503
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

800 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)(9,10,14,18)(11,20,21,16)(12,22,23,24)(13,19,15,17)(25,26)(27,34)(28,38)(29,33)(30,32)(31,37)(35,40)(36,39);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,13)(14,21)(15,23)(17,22)(18,20)(19,24)(25,27,29,37)(26,31,33,34)(28,36,30,35)(32,39,38,40);(1,4,5,3)(2,6,7,8)(9,12,14,23)(10,16,18,20)(11,13,21,15)(17,22,19,24)(27,35)(28,30)(31,39)(32,38)(34,40)(36,37);(2,7)(6,8)(9,13,14,15)(10,17,18,19)(11,12,21,23)(16,22,20,24)(25,28,29,30)(26,32,33,38)(27,36,37,35)(31,39,34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 163915);