#93607

order	512
nilpotency class	4
isoclinism	isoclinic class of #93565
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1286
commutator	C₂×C₄²	C₄²
Frattini, cosocle	#1726	C₂²
socle	C₂	#7040
prime radical	C₂³	#344
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ #1726 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

161 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 161 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=fi,g^2=i,b^a=bc,c^a=cf,e^a=efgh,f^a=fi,h^a=hi,c^b=cg,d^b=dh,h^b=hi,d^c=di,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, 803);