#138783

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	C₂³	#386
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#557 ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

411 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 411 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=i,d^2=gh,e^2=ghi,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=egh,f^b=fh,g^b=gi,h^b=hi,d^c=dgh,e^c=eghi,f^c=fgh,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, 45979);