#150928

order	512
nilpotency class	5
isoclinism	isoclinic class of #150407
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	C₂×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	#382	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#124 ◁ G
LECS	G ▷ #382 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

355 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 24 points:

(2,6)(3,4)(7,8)(10,14,16,23,17,19,15,24)(11,18,20,21);(1,2,5,8)(3,6,4,7)(9,10,12,15,13,17,22,16)(11,19,18,24,20,14,21,23);(1,3,5,4)(2,7,8,6)(10,15,17,16)(14,23,19,24)

Polycyclic presentation:

2^9|a^2=e,b^2=fh,c^2=g,d^2=gh,e^2=h,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=cf,d^b=dg,e^b=efghi,f^b=fi,g^b=gi,h^b=hi,d^c=dh,e^d=ehi,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58124);