#256740

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

771 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)(9,10,14,19)(11,17,22,16)(12,23,21,24)(13,20,15,18)(25,26)(27,35)(28,34)(29,33)(30,32)(31,38)(36,40)(37,39);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,15)(13,21)(14,22)(17,19)(18,24)(20,23)(25,27)(26,31)(28,37)(29,38)(30,36)(32,39)(33,35)(34,40);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,13)(14,21)(15,22)(16,19)(18,23)(20,24)(26,32)(27,36)(28,30)(33,34)(37,38)(39,40);(2,7)(6,8)(9,13,14,15)(10,18,19,20)(11,21,22,12)(16,24,17,23)(25,28,29,30)(26,32,33,34)(27,37,38,36)(31,39,35,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,c^2=eh,d^2=h,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, 163936);