#94547

order	512
nilpotency class	6
isoclinism	isoclinic class of #94547
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7345
commutator	#125	C₂×C₈
Frattini, cosocle	#1991	C₂²
prime radical	#3553	C₄
LCS	G ▷ #125 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1991 ◁ G
LECS	G ▷ #1991 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

198 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 26 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)(3,14,16,45,17,35,23,58)(4,19,20,55)(6,11,29,40)(7,13,25,41)(8,31,32,50,34,51,61,53)(9,24,36,22)(12,26,44,30)(15,42,48,62,49,33,59,38)(18,46,54,60,27,47,52,63)(21,43,57,39)(28,37,56,64);(1,3)(2,8)(4,15)(5,23)(6,27)(7,17)(9,35)(10,38)(11,42)(12,33)(13,34)(14,37)(16,25)(18,30)(19,46)(20,48)(21,49)(22,53)(24,31)(26,52)(28,50)(29,54)(32,40)(36,63)(39,45)(41,62)(43,47)(44,61)(51,56)(55,58)(57,59)(60,64)

Polycyclic presentation:

2^9|a^2=d,c^2=h,d^2=f,e^2=h,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ch,d^b=de,e^b=eh,f^b=fi,g^b=ghi,d^c=dghi,e^c=ehi,g^c=gi,e^d=ehi,g^d=gi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1743);