#152147

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	#3221	C₂²
LCS	G ▷ #117 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ #3509 ◁ G
LECS	G ▷ #536 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 531 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)(3,14,18,45)(4,20,23,12)(5,24,26,54)(6,11)(7,13)(8,16,32,47)(9,22,36,58)(10,21)(15,46,19,38)(17,44,53,43)(25,39,28,35)(27,59,48,61)(29,37)(30,60,31,49)(33,50,34,51)(40,55)(41,56)(42,57)(52,62,64,63);(1,3)(2,8)(4,15)(5,16)(6,27)(7,19)(9,30)(10,38)(11,31)(12,32)(13,34)(14,41)(17,29)(18,23)(20,33)(21,47)(22,53)(24,46)(25,52)(26,51)(28,49)(35,43)(36,62)(37,63)(39,61)(40,59)(42,44)(45,54)(48,58)(50,56)(55,64)(57,60);(1,4,7,23)(2,9)(3,15,19,18)(5,21,26,56)(6,22,29,58)(8,30)(10,39)(11,20)(12,37)(13,36)(14,43)(16,47,51,50)(17,48,27,53)(24,42)(25,55,28,57)(31,33)(32,63)(34,62)(35,41)(38,61)(40,54)(44,46)(45,59)(49,60,52,64)

Polycyclic presentation:

2^9|a^2=f,c^2=hi,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, 59343);