#123264

order	512
nilpotency class	4
isoclinism	isoclinic class of #123243
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
commutator, Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
prime radical	#1999	C₂²
LCS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

365 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=fh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=egi,f^b=fh,d^c=dg,e^c=ehi,e^d=ei

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30460);