#336906

order	512
nilpotency class	3
isoclinism	isoclinic class of #332071
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

566 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=e,d^2=hi,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, 244102);