#139813

order	512
nilpotency class	4
isoclinism	isoclinic class of #139812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9497
commutator	C₂²×Q₈	C₂²×C₄
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	#3486	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

643 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 643 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=h,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,h^a=hi,d^b=dgi,e^b=ehi,f^b=fgi,g^b=gi,d^c=dh,f^c=fh,e^d=ei,f^d=fi,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, 47009);