#152911

order	512
nilpotency class	5
isoclinism	isoclinic class of #149893
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2122
commutator	C₂×#41	C₂²×C₄
Frattini, cosocle	#532	C₂³
socle	C₂	#13218
prime radical	#24287	C₂
LCS	G ▷ C₂×#41 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 453 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=e,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,d^b=dg,f^b=fg,g^b=gh,f^d=fhi,g^d=gi,g^f=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60107);