#150040

order	512
nilpotency class	5
isoclinism	isoclinic class of #149901
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×AΣL(1,𝔽₁₆)
commutator	C₂²×Q₈	C₂²×C₄
Frattini, cosocle	#391	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂²×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#124 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

340 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=fh,c^2=gh,d^2=i,e^2=g,f^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,c^b=cgh,d^b=dhi,e^b=ef,f^b=fi,g^b=gi,h^b=hi,e^d=ehi,f^d=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57236);