#382473

order	512
nilpotency class	3
isoclinism	isoclinic class of #382472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14336
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2221 conjugacy classes of subgroups; only the 352 normal subgroups are shown.

normal subgroups only

This group has 2221 conjugacy classes of subgroups — too many to draw. Only its 352 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,h^d=hi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289669);