#153284

order	512
nilpotency class	5
isoclinism	isoclinic class of #153131
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2199
commutator, Frattini, cosocle	#532	C₂³
socle	C₂	#13419
prime radical	#31077	C₂
LCS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

418 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 46 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)(3,15)(4,19)(5,24)(6,16)(7,26)(8,12)(9,14)(10,17)(11,22)(13,25)(18,27)(20,21)(23,30)(28,32)(29,31);(1,3)(2,10)(4,14)(5,19)(6,11)(7,18)(8,17)(9,20)(12,21)(13,27)(15,24)(16,23)(22,26)(25,31)(28,30)(29,32);(1,4,5,21,9,10,8,15)(2,11,12,30,14,31,24,18)(3,16,17,32,20,13,19,26)(6,22,7,27,25,29,28,23)

Polycyclic presentation:

2^9|c^2=d,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fghi,h^a=hi,c^b=cf,d^b=di,e^b=eg,g^b=gi,h^b=hi,f^c=fi,g^c=gh,f^e=fhi,g^e=gi,g^f=gi

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60480);