#335515

order	512
nilpotency class	3
isoclinism	isoclinic class of #332071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

694 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=ei,c^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=deh,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 242711);