#123373

order	512
nilpotency class	4
isoclinism	isoclinic class of #123363
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1667
commutator, Frattini, cosocle	C₈²	C₂³
prime radical	C₂×D₈	D₈
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

160 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 52 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)(3,10)(4,11)(7,12)(8,16)(9,15)(13,14)(17,18,21,25)(19,22,28,31)(20,29,27,24)(23,32,30,26);(1,2,7,16,4,8,3,6,5,9,10,14,11,13,12,15);(18,22)(19,26)(20,27)(23,24)(25,31)(28,32)(29,30)

Polycyclic presentation:

2^9|a^2=i,b^2=dfh,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,e^c=eg,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226835, #72226921, #72226932
maximal normal subgroup of	#72226835, #72226921, #72226932
maximal quotient of	#72226833

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30569);