#138515

order	512
nilpotency class	4
isoclinism	isoclinic class of #138482
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#29730	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

528 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=i,d^2=ghi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eg,f^b=fh,h^b=hi,d^c=dg,f^c=fg

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, 45711);