#152943

order	512
nilpotency class	5
isoclinism	isoclinic class of #152939
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13223
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#532	C₂³
prime radical	#28497	C₂
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 405 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=ei,d^2=hi,e^2=i,f^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=dgh,f^b=fgh,g^b=ghi,h^b=hi,f^d=fhi,g^f=gi

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, 60139);