#100230

order	512
nilpotency class	3
isoclinism	isoclinic class of #100227
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

664 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 86 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)(3,7,8,5)(4,6)(9,10,13,15,12,11,14,16)(18,22)(19,26)(20,23)(25,30)(27,28)(29,31);(1,2)(3,5)(4,6)(7,8)(9,11,13,16,12,10,14,15)(17,18,23,30,21,25,20,22)(19,24,26,31,28,32,27,29);(1,3)(2,5)(4,8)(6,7)(10,11)(15,16)(17,19,21,28)(18,24,25,32)(20,27,23,26)(22,29,30,31)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=g,d^2=g,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,c^b=ci,e^b=eh,d^c=dh,e^c=eg,f^c=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7426);