#123365

order	512
nilpotency class	4
isoclinism	isoclinic class of #123363
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

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

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

normal subgroups only

This group has 169 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,6)(4,13)(5,9)(7,10)(8,11)(12,15)(14,16)(17,18,22,26)(19,27,31,43)(20,32,34,47)(21,35,33,25)(23,36,38,29)(24,39,40,28)(30,46,45,42)(37,44,41,48);(17,19,28,44,21,30,34,36,22,31,39,48,33,45,20,29)(18,23,32,42,25,37,40,43,26,38,47,46,35,41,24,27);(2,6)(3,10)(4,11)(7,8)(9,15)(12,16)(13,14)

Polycyclic presentation:

2^9|a^2=h,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,e^c=eg,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225448, #72227078
maximal normal subgroup of	#72225448, #72227078
maximal quotient of	#72225443

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30561);