#94649

order	512
nilpotency class	6
isoclinism	isoclinic class of #94648
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7403
commutator	C₂×#97	C₂×C₄
Frattini, cosocle	#1960	C₂²
prime radical	C₂³	#353
LCS	G ▷ C₂×#97 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1960 ◁ G
LECS	G ▷ #1960 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

159 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 159 conjugacy classes of subgroups — too many to draw. Only its 24 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=h,c^2=f,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fh,g^a=gh,h^a=hi,c^b=cf,d^b=defh,e^b=ehi,f^b=fi,d^c=dgh,e^c=ei,e^d=ehi,f^d=fi,g^d=gi

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1845);