#94448

order	512
nilpotency class	5
isoclinism	isoclinic class of #94447
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7310
commutator	C₂²×D₄	C₄²
Frattini, prime radical, cosocle	#3487	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ #3487 ◁ G
LECS	G ▷ #3487 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

359 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 359 conjugacy classes of subgroups — too many to draw. Only its 38 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)(3,14,16,44)(4,19,20,53)(6,26,24,41)(7,12,29,43)(8,13,25,42)(9,30,31,48)(10,34,36,28)(15,32,47,39)(17,46,23,57)(18,35,52,60)(21,38,55,40)(22,37,56,64)(27,45,51,63)(33,54,62,49)(50,58,59,61);(1,3,6,17,8,18,7,27)(2,9,12,32,13,33,26,58)(4,15,21,48,22,50,28,49)(5,23,24,52,25,51,29,16)(10,35,37,46,38,14,19,45)(11,39,41,31,42,61,43,62)(20,54,55,47,56,30,34,59)(36,63,53,44,40,57,64,60)

Polycyclic presentation:

2^9|a^2=d,b^2=e,e^2=i,f^2=i,b^a=bc,c^a=cf,e^a=eg,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,d^b=dfi,f^b=fh,d^c=dhi,f^c=fi,e^d=ehi,f^d=fi,g^d=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72266386

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1644);