#505684

order	512
nilpotency class	4
isoclinism	isoclinic class of #500307
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3227
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#27645
prime radical	#3054	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

541 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 541 conjugacy classes of subgroups — too many to draw. Only its 152 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:

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

Polycyclic presentation:

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

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, 412880);