#149768

order	512
nilpotency class	5
isoclinism	isoclinic class of #149765
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11503
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	#407	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2920 ◁ G
LECS	G ▷ #407 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 406 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=hi,e^2=g,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ch,e^b=ef,g^b=gi,e^c=ei,e^d=eh,f^e=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56964);