#124816

order	512
nilpotency class	4
isoclinism	isoclinic class of #124690
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1548
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
socle	C₂	#7624
prime radical	C₂³	#410
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₄³ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 261 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)(3,12,15,7,14,21,19,22)(4,8)(5,9)(6,11)(10,16)(13,27,28,32,29,26,31,20)(17,24)(18,25)(23,30);(1,3,6,14)(2,7)(4,13,17,29)(5,19,16,15)(8,20)(9,12)(10,21)(11,22)(18,31,30,28)(23,27)(24,32)(25,26);(1,4,16,30,6,17,5,18)(2,8,10,23,11,24,9,25)(3,13,15,28,14,29,19,31)(7,20,21,27,22,32,12,26)

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=fgh,d^2=gi,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=efg,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, 32012);