#94446

order	512
nilpotency class	5
isoclinism	isoclinic class of #94445
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×Q₈	C₄²
Frattini, prime radical, cosocle	#3485	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ #3485 ◁ G
LECS	G ▷ #3485 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 362 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)(4,19,21,54)(6,27,25,41)(7,12,30,43)(8,13,26,42)(9,31,32,51)(10,35,36,29)(15,48,49,39)(17,46,52,64)(18,47,53,60)(20,33,55,62)(22,38,56,40)(23,37,57,63)(24,58,28,45)(34,59,61,50);(1,3,6,17,8,18,7,28)(2,9,12,33,13,34,27,48)(4,20,22,51,23,15,29,50)(5,24,25,16,26,52,30,53)(10,14,37,45,38,47,19,46)(11,39,41,61,42,62,43,32)(21,31,56,49,57,59,35,55)(36,58,54,44,40,64,63,60)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,e^2=i,f^2=i,b^a=bc,c^a=cf,e^a=egi,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,d^b=df,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

#72266409

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1642);