#123149

order	512
nilpotency class	4
isoclinism	isoclinic class of #123131
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

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

316 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30345);