#152152

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, prime radical, cosocle	#535	C₂³
LCS	G ▷ #117 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ #3510 ◁ G
LECS	G ▷ #535 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 538 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=h,e^2=i,f^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, 59348);