#149946

order	512
nilpotency class	5
isoclinism	isoclinic class of #149945
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11557
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	#407	C₂³
prime radical	#24281	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2929 ◁ G
LECS	G ▷ #407 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

459 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 459 conjugacy classes of subgroups — too many to draw. Only its 54 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=gi,e^2=g,g^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cg,e^b=ef,g^b=gi,e^d=eh,f^e=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, 57142);