#148489

order	512
nilpotency class	4
isoclinism	isoclinic class of #148489
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11174
commutator	C₂²×Q₈	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

800 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 120 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,e^c=ehi,g^c=ghi,h^c=hi,e^d=ei,g^d=gi,f^e=fi,g^e=gi,g^f=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55685);