#507942

order	512
nilpotency class	4
isoclinism	isoclinic class of #500591
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3253
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#29533
prime radical	#3414	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 462 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)(3,13)(4,8)(5,19)(6,10)(7,12)(9,15)(11,18)(14,27)(16,30)(17,29)(20,26)(21,28)(22,24)(23,31)(25,32);(2,8,12,10)(3,14,17,16)(9,25)(11,24,28,19)(13,29)(15,31,32,20)(23,26)(27,30);(1,3)(2,9)(4,14)(5,20)(6,16)(7,17)(8,23)(10,26)(11,27)(12,25)(13,19)(15,18)(21,32)(22,31)(24,29)(28,30);(2,8,12,10)(9,23,25,26)(11,24,28,19)(13,27,29,30)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415138);