#511761

order	512
nilpotency class	4
isoclinism	isoclinic class of #511761
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32878
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3100 conjugacy classes of subgroups; only the 220 normal subgroups are shown.

normal subgroups only

This group has 3100 conjugacy classes of subgroups — too many to draw. Only its 220 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418957);