#93649

order	512
nilpotency class	4
isoclinism	isoclinic class of #93553
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1300
commutator	C₂×C₄²	C₄²
Frattini, cosocle	#1726	C₂²
socle	C₂	#7048
prime radical	C₂³	#344
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ #1726 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

161 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 161 conjugacy classes of subgroups — too many to draw. Only its 48 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=gh,e^2=h,f^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efgh,f^a=fi,c^b=cg,d^b=dh,g^b=gi,e^c=ei,e^d=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, 845);