#312159

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

555 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 219355);