#150474

order	512
nilpotency class	5
isoclinism	isoclinic class of #150474
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11773
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	#408	C₂³
prime radical	#525	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2929 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

273 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 273 conjugacy classes of subgroups — too many to draw. Only its 54 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=g,d^2=g,g^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cf,d^b=dg,e^b=efgi,g^b=gi,d^c=dh,f^c=fi,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, 57670);