#94397

order	512
nilpotency class	5
isoclinism	isoclinic class of #7396
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#353
commutator	C₂×M₁₆	C₄²
Frattini, cosocle	#2929	C₂²
socle	C₂	#7287
prime radical	#525	Q₈
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2929 ◁ G
LECS	G ▷ #2929 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

180 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 42 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fi,d^2=fh,e^2=fh,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gh,e^c=ehi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1593);