#151365

order	512
nilpotency class	5
isoclinism	isoclinic class of #151364
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12195
commutator, Frattini, cosocle	#423	C₂³
prime radical	#575	D₄
LCS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #423 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

263 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 263 conjugacy classes of subgroups — too many to draw. Only its 36 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fgh,d^2=fi,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=ehi,f^a=fi,g^a=gh,d^b=df,e^b=eghi,f^b=fi,h^b=hi,d^c=dg,e^d=eh,g^d=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, 58561);