#328776

order	512
nilpotency class	3
isoclinism	isoclinic class of #327021
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

759 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,17,18,24)(13,21,20,16)(15,22,23,19)(27,35)(28,29)(31,39)(33,34)(36,40)(37,38);(2,5)(6,8)(10,14)(15,23)(16,21)(17,24)(25,26,30,32)(27,36,38,39)(28,34,29,33)(31,35,40,37);(3,7)(6,8)(9,11,12,18)(10,15,14,23)(13,22,20,19)(16,24,21,17)(25,27,30,38)(26,31,32,40)(28,37,29,35)(33,36,34,39);(1,3,4,7)(2,6,5,8)(11,18)(15,23)(17,24)(19,22)(26,32)(31,40)(33,34)(36,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=di,f^b=fi,g^b=gi,d^c=de,f^c=fh,g^c=gi

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, 235972);