#151956

order	512
nilpotency class	5
isoclinism	isoclinic class of #13605
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2127
commutator	C₄²	C₂×C₄²
Frattini, cosocle	#532	C₂³
socle	C₂	#12628
prime radical	#3150	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2932 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

341 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 341 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eh,d^2=h,e^2=i,g^2=i,b^a=bd,c^a=ch,d^a=dhi,f^a=fgh,g^a=gi,d^b=dg,e^b=ei,g^b=gh,h^b=hi,d^c=di,f^d=fhi,g^f=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59152);