#6432152

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432152
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#396
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3053 conjugacy classes of subgroups; only the 904 normal subgroups are shown.

normal subgroups only

This group has 3053 conjugacy classes of subgroups — too many to draw. Only its 904 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 — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,f^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,h^b=hi,f^c=fi,e^d=ei

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339348);