#256535

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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

666 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)(9,10,15,19)(11,12,21,23)(13,17,14,18)(16,24,20,22)(25,26)(27,32)(28,37)(29,33)(30,38)(31,34)(35,39)(36,40);(1,3,5,4)(2,6,7,8)(9,11)(10,12)(13,20)(14,16)(15,21)(17,22)(18,24)(19,23)(25,27)(26,32)(28,30)(29,35)(31,36)(33,39)(34,40)(37,38);(1,4,5,3)(2,6,7,8)(9,12)(10,16)(11,17)(13,22)(14,24)(15,23)(18,21)(19,20)(25,28)(26,32)(27,29)(30,36)(31,35)(33,39)(34,40)(37,38);(2,7)(6,8)(9,13,15,14)(10,17,19,18)(11,20,21,16)(12,22,23,24)(25,29,31,30)(26,33,34,38)(27,35,36,28)(32,39,40,37)

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=e,d^2=h,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, 163731);