#128199

order	512
nilpotency class	4
isoclinism	isoclinic class of #128057
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#342
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

640 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 138 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)(3,11,13,7)(4,8)(5,10)(6,9)(12,16,14,15)(18,22,24,29)(19,25)(23,30,31,28)(26,32);(1,3,5,14,6,13,4,12)(2,7,10,16,9,11,8,15)(17,18)(19,23,25,31)(20,22,27,29)(21,24)(26,28)(30,32);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,31)(27,32)(29,30)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=i,f^2=g,b^a=bd,c^a=ci,d^a=dg,f^a=fhi,c^b=ch,d^b=dh,e^b=egi,h^b=hi,f^c=fi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35395);