#256521

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

715 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)(11,12)(13,17)(14,18)(15,19)(16,22)(20,24)(21,23)(25,26)(27,33)(28,39)(29,36)(30,35)(31,34)(32,38)(37,40);(1,3,5,4)(2,6,7,8)(9,11,15,21)(10,12,19,23)(13,16,14,20)(17,22,18,24)(25,27)(26,32)(28,31)(29,37)(30,38)(33,35)(34,40)(36,39);(1,4,5,3)(2,6,7,8)(9,12)(10,16)(11,18)(13,22)(14,24)(15,23)(17,21)(19,20)(25,28)(26,33)(27,29)(30,37)(31,38)(32,35)(34,39)(36,40);(2,7)(6,8)(9,13,15,14)(10,17,19,18)(11,16,21,20)(12,22,23,24)(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=ei,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, 163717);