#324439

order	512
nilpotency class	3
isoclinism	isoclinic class of #278155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
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

674 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)(3,7)(4,5)(6,8)(17,18)(19,25)(20,22)(21,29)(23,27)(24,28)(26,31)(30,32);(2,5)(6,7)(9,10,13,14,12,16,11,15)(18,22)(23,30)(24,29)(25,31);(10,14)(11,13)(15,16)(17,19,21,27,20,26,28,32)(18,23,24,25,22,30,29,31);(1,3,4,8)(2,6,5,7)(18,22)(19,26)(24,29)(27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 231635);