#152122

order	512
nilpotency class	5
isoclinism	isoclinic class of #152121
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12697
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#124	C₂³
prime radical	#60828	C₂
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #117 ◁ #3510 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

629 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 629 conjugacy classes of subgroups — too many to draw. Only its 76 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)(3,13,17,39)(4,19)(5,24)(6,11,28,36)(7,12)(8,15,31,44)(9,22)(10,26,34,57)(14,37,42,61)(16,38,18,40)(20,49,43,62)(21,51)(23,48,54,33)(25,35,45,30)(27,56)(29,41,58,63)(32,59,47,53)(46,50,60,64)(52,55);(1,3)(2,8)(4,20)(5,25)(6,17)(7,16)(9,29)(10,13)(11,35)(12,30)(14,22)(15,27)(18,28)(19,46)(21,41)(23,43)(24,39)(26,45)(31,36)(32,37)(33,60)(34,38)(40,56)(42,54)(44,57)(47,49)(48,58)(50,52)(51,62)(53,63)(55,61)(59,64);(1,4)(2,9,12,19)(3,14)(5,21)(6,23)(7,22)(8,29,30,46)(10,32,34,47)(11,33,36,48)(13,37,38,49)(15,41)(16,20)(17,42)(18,43)(24,55,56,51)(25,50)(26,53)(27,52)(28,54)(31,58,35,60)(39,61,40,62)(44,63)(45,64)(57,59)

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fghi,g^b=gi,e^c=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, 59318);