#508490

order	512
nilpotency class	4
isoclinism	isoclinic class of #508490
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#29890
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#32945	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

598 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 128 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)(4,9)(5,11)(6,10)(7,12)(8,15)(14,28)(16,24)(17,30)(18,26)(19,32)(20,29)(21,27)(22,25)(23,31);(1,3,4,16,7,15,6,22)(2,8,10,24,12,13,9,25)(5,19,18,31,20,14,21,17)(11,28,27,23,29,32,26,30);(2,9)(3,14)(4,6)(5,18)(8,23)(10,12)(11,29)(13,30)(15,19)(16,31)(17,22)(20,21)(24,28)(25,32);(5,20)(11,29)(14,19)(17,31)(18,21)(23,30)(26,27)(28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415686);