#128925

order	512
nilpotency class	4
isoclinism	isoclinic class of #11994
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1558
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8232
prime radical	C₂³	#413
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

282 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36121);