#149958

order	512
nilpotency class	5
isoclinism	isoclinic class of #149958
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11560
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	#407	C₂³
prime radical	#3003	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2930 ◁ G
LECS	G ▷ #407 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

315 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 315 conjugacy classes of subgroups — too many to draw. Only its 52 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=g,e^2=g,g^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cgh,e^b=ef,g^b=gi,d^c=di,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, 57154);