#138830

order	512
nilpotency class	4
isoclinism	isoclinic class of #138776
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂×#493	C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#559 ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

494 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,d^2=ghi,e^2=gh,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,d^b=dh,e^b=eghi,f^b=fhi,g^b=gi,h^b=hi,d^c=dghi,e^c=egh,f^c=fghi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46026);