#123183

order	512
nilpotency class	4
isoclinism	isoclinic class of #123171
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×C₄⋊C₈	C₂³
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 348 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)(9,10,15,21)(11,22,25,37)(12,16,29,33)(13,20,14,19)(17,30,35,24)(18,28,32,26)(23,36,27,34)(31,39,38,40);(3,8)(10,16)(12,26)(13,14)(17,31)(18,20)(19,32)(21,33)(22,36)(23,39)(24,30)(27,40)(28,29)(34,37)(35,38);(2,5,7,6)(9,11)(10,17)(12,27)(13,30)(14,24)(15,25)(16,31)(18,36)(19,37)(20,22)(21,35)(23,29)(26,40)(28,39)(32,34)(33,38)

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30379);