#94807

order	512
nilpotency class	6
isoclinism	isoclinic class of #94807
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7504
commutator	C₄×C₈	C₄²
Frattini, prime radical, cosocle	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #3147 ◁ G
LECS	G ▷ #3147 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

188 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 36 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)(3,15,17,45)(4,10,20,36)(6,13,28,44)(7,12,27,42)(8,14,26,43)(9,30,31,49)(16,34,48,40)(18,46,53,62)(19,35,24,58)(21,38,25,41)(22,37,56,63)(23,39,55,64)(29,47,52,59)(32,54,60,50)(33,51,61,57);(1,3)(2,9,12,32,14,33,13,34)(4,16)(5,24,26,17)(6,29)(7,18)(8,19)(10,35,37,47,39,15,38,46)(11,40,42,31,43,60,44,61)(20,54,55,48)(21,57)(22,49)(23,50)(25,30,56,51)(27,52,28,53)(36,62,63,58,64,59,41,45)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=gh,e^2=h,f^2=hi,g^2=i,b^a=bc,c^a=cf,e^a=eg,f^a=fg,g^a=gh,h^a=hi,c^b=ch,d^b=dfg,f^b=fi,d^c=dgh,e^d=ehi,f^d=fhi,g^d=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2003);