#134826

order	512
nilpotency class	4
isoclinism	isoclinic class of #134781
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 338 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=i,d^2=g,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cgi,f^c=fh,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, 42022);