#139746

order	512
nilpotency class	4
isoclinism	isoclinic class of #139745
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9495
commutator	C₂²×Q₈	C₂²×C₄
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	#23899	C₂
LCS	G ▷ C₂²×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 659 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,d^b=dgi,e^b=ehi,f^b=fgi,g^b=gi,h^b=hi,d^c=dh,e^c=ei,f^c=fh,g^c=gi,e^d=ei,f^d=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, 46942);