#134831

order	512
nilpotency class	4
isoclinism	isoclinic class of #134787
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1799
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂²×D₄	C₄⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

320 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,d^2=g,e^2=i,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cg,e^c=ei,f^c=fhi,f^e=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, 42027);