#152146

order	512
nilpotency class	5
isoclinism	isoclinic class of #152145
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12708
commutator	#117	C₂²×C₄
Frattini, cosocle	#536	C₂³
prime radical	#535	D₄
LCS	G ▷ #117 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ #3510 ◁ G
LECS	G ▷ #536 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

532 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 74 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=h,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dh,e^b=ehi,f^b=fghi,g^b=gi,d^c=dh,f^c=fi,e^d=ei,f^d=fhi,g^f=gi

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59342);