#94637

order	512
nilpotency class	6
isoclinism	isoclinic class of #94636
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7402
commutator	C₂×#97	C₂×C₄
Frattini, cosocle	#1960	C₂²
prime radical	#3150	C₄
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

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

normal subgroups only

This group has 186 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:

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

Polycyclic presentation:

2^9|a^2=d,c^2=fi,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=defhi,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, 1833);