#152154

order	512
nilpotency class	5
isoclinism	isoclinic class of #152148
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	#535	C₂³
prime radical	#3220	C₂²
LCS	G ▷ #117 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ #3510 ◁ G
LECS	G ▷ #535 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 545 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)(3,16,19,44,21,36,53,34)(4,22,24,40,26,45,29,10)(7,11,28,42,32,14,30,43)(9,27,35,20,37,18,47,31)(17,48,50,38,23,57,52,33)(25,55,61,41,62,56,60,39)(46,59,64,49,63,51,54,58);(1,3)(2,9)(4,23)(5,27)(6,18)(7,31)(8,21)(10,38)(11,36)(12,34)(13,44)(14,16)(15,37)(17,26)(19,28)(20,32)(22,54)(24,49)(25,51)(29,58)(30,53)(33,40)(35,42)(39,63)(41,46)(43,47)(45,64)(48,55)(50,60)(52,61)(56,57)(59,62);(1,4)(2,10,15,40)(3,17)(5,24)(6,29)(7,25)(8,26)(9,33,37,38)(11,41,14,39)(12,22,13,45)(16,46,36,63)(18,49)(19,52)(20,51)(21,23)(27,58)(28,60)(30,61)(31,59)(32,62)(34,64,44,54)(35,57,47,48)(42,55,43,56)(50,53)

Polycyclic presentation:

2^9|a^2=f,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,c^b=ci,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

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59350);