#309107

order	512
nilpotency class	3
isoclinism	isoclinic class of #264203
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

561 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=fi,c^2=fh,d^2=ei,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 216303);