#256631

order	512
nilpotency class	3
isoclinism	isoclinic class of #246415
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

815 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)(25,26,28,32)(27,34,36,30)(29,31,38,33)(35,40,37,39);(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)(26,30)(28,36)(29,35)(31,39)(32,34)(33,40)(37,38);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,13)(14,21)(15,22)(16,19)(18,23)(20,24)(26,31)(27,35)(29,38)(32,33)(36,37)(39,40);(2,7)(6,8)(9,13,14,15)(10,18,19,20)(11,21,22,12)(16,24,17,23)(26,32)(30,34)(31,33)(39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=eh,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,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 163827);