#488120

order	512
nilpotency class	4
isoclinism	isoclinic class of #487619
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#43360	C₂
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

930 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 118 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=gi,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,e^b=eg,f^b=fhi,d^c=dg,e^c=eh,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234734, #72234737, #72234738
maximal normal subgroup of	#72234734, #72234737, #72234738

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395316);