#123383

order	512
nilpotency class	4
isoclinism	isoclinic class of #123383
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1667
commutator, Frattini, cosocle	C₈²	C₂³
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

193 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 193 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 256

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

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

Polycyclic presentation:

2^9|b^2=df,d^2=fh,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,g^a=gi,c^b=ch,e^c=eg,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226856, #72226920, #72226932
maximal normal subgroup of	#72226856, #72226920, #72226932
maximal quotient of	#72226853

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30579);