#330242

order	512
nilpotency class	3
isoclinism	isoclinic class of #329515
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

679 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)(3,6)(4,5)(7,8)(9,10,12,13)(11,15,16,14)(17,18,22,30,21,24,20,25)(19,26,29,23,28,32,27,31);(2,5)(6,8)(10,13)(11,14)(15,16);(3,7)(6,8)(9,11)(10,14)(12,16)(13,15)(17,19,21,28)(18,23,24,31)(20,29,22,27)(25,32,30,26);(1,3,4,7)(2,6,5,8)(11,16)(14,15)(18,24)(23,31)(25,30)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 237438);