#152129

order	512
nilpotency class	5
isoclinism	isoclinic class of #149800
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2129
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#532	C₂³
socle	C₂	#12698
prime radical	#3455	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3455 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 534 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)(3,14,18,31)(4,19,21,48)(5,11)(6,12,27,10)(7,13)(8,16,30,26)(9,22,33,51)(15,40,42,46)(17,39,45,59)(20,49,43,63)(23,47)(24,36,55,35)(25,37)(28,50,38,62)(29,44,58,54)(32,60)(34,53)(41,56,64,57)(52,61);(1,3,7,16)(2,8)(4,20,21,15)(5,17,25,44)(6,26,27,18)(9,28)(10,14)(11,29)(12,30)(13,31)(19,46)(22,50,51,41)(23,43,52,42)(24,54,55,45)(32,56)(33,57)(34,38)(35,39)(36,58)(37,59)(40,61)(47,49)(48,63)(53,62,60,64);(1,4,7,21)(2,9)(3,15,16,20)(5,22,25,51)(6,23,27,52)(8,28)(10,34)(11,19)(12,32)(13,33)(14,38)(17,41,44,50)(18,43,26,42)(24,53,55,60)(29,46)(30,56)(31,57)(35,61)(36,47)(37,48)(39,40)(45,62,54,64)(49,58)(59,63)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=h,d^2=hi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ch,d^b=dh,f^b=fgh,g^b=gi,d^c=di,f^c=fi,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, 59325);