#124794

order	512
nilpotency class	4
isoclinism	isoclinic class of #124661
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1545
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
socle	C₂	#7621
prime radical	#3150	C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₄³ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

347 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 76 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)(3,12,14,22,15,21,18,7)(4,8)(5,9)(6,11)(10,19)(13,27,29,20,28,26,31,32)(16,24)(17,25)(23,30);(1,3)(2,7)(4,13)(5,18)(6,15)(8,20)(9,21)(10,12)(11,22)(14,19)(16,28)(17,31)(23,26)(24,32)(25,27)(29,30);(1,4,5,17,6,16,19,30)(2,8,9,25,11,24,10,23)(3,13,18,31,15,28,14,29)(7,20,21,27,22,32,12,26)

Polycyclic presentation:

2^9|a^2=e,c^2=fgh,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,d^c=di

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, 31990);