#337776

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

592 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)(3,5)(4,6)(7,8)(9,10,14,18)(11,15,13,16)(12,20,21,23)(17,22,24,19)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,3,5,2)(4,7,8,6)(9,11,14,13)(10,15,18,16)(12,19,21,22)(17,23,24,20)(25,27)(26,30)(28,35)(29,34)(31,33)(32,38)(36,37)(39,40);(1,4,5,8)(2,6,3,7)(10,16)(11,13)(15,18)(17,20)(19,22)(23,24)(27,34)(29,37)(30,38)(32,40)(33,39)(35,36);(2,3)(6,7)(9,12,14,21)(10,17,18,24)(11,19,13,22)(15,23,16,20)(26,31)(27,35)(32,40)(34,36)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=ei,d^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=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 244972);