#128893

order	512
nilpotency class	4
isoclinism	isoclinic class of #11912
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1530
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×#97	C₂³
socle	C₂	#8226
prime radical	#532	C₂×C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

310 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=gi,e^2=i,f^2=h,h^2=i,b^a=bd,d^a=dg,f^a=fghi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,g^b=gi,d^c=di,f^c=fi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36089);