#505574

order	512
nilpotency class	4
isoclinism	isoclinic class of #499952
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3205
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#27582
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

605 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 605 conjugacy classes of subgroups — too many to draw. Only its 134 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412770);