#150218

order	512
nilpotency class	5
isoclinism	isoclinic class of #150216
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11692
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	#408	C₂³
prime radical	#29751	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #2879 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

368 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=e,d^2=gi,e^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=ch,d^b=dg,e^b=efg,g^b=gi,d^c=di,e^d=eh,f^e=fi

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, 57414);