#93554

order	512
nilpotency class	4
isoclinism	isoclinic class of #93553
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1300
commutator	C₂×C₄²	C₄²
Frattini, cosocle	C₂×#405	C₂²
socle	C₂	#7036
prime radical	C₂⁴	#101
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#405 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

246 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 64 points:

(1,2,5,10,6,11,27,42,7,13,28,44,29,12,45,43)(3,14,16,46,18,47,25,59,19,36,54,61,17,48,53,64)(4,20,21,56,23,39,58,41,24,38,26,60,22,9,35,37)(8,30,31,15,33,49,62,50,34,57,63,51,32,52,40,55);(1,3,6,17,7,19,29,18)(2,8,11,32,13,34,12,33)(4,15,22,50,24,51,23,55)(5,25,27,16,28,53,45,54)(9,36,38,48,39,14,20,47)(10,40,42,63,44,62,43,31)(21,57,35,52,26,30,58,49)(37,59,60,61,41,64,56,46)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efgh,g^a=gi,h^a=hi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,h^b=hi,d^c=di,e^c=ei

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, 750);