#123080

order	512
nilpotency class	4
isoclinism	isoclinic class of #123071
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator, Frattini, cosocle	C₂×C₄⋊C₈	C₂³
prime radical	#3245	C₂²
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

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

normal subgroups only

This group has 303 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) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,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,c^b=ci,d^b=df,e^b=egi,f^b=fh,d^c=dg,e^c=ei,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, 30276);