#153122

order	512
nilpotency class	5
isoclinism	isoclinic class of #153121
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13343
commutator, Frattini, cosocle	#532	C₂³
prime radical	#31077	C₂
LCS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

577 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 577 conjugacy classes of subgroups — too many to draw. Only its 50 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)(3,8)(4,18)(5,23)(6,12)(7,13)(9,20)(10,27)(11,29)(14,43)(15,39)(16,33)(17,34)(19,51)(21,52)(22,40)(24,31)(25,38)(26,41)(28,32)(30,45)(35,62)(36,53)(37,54)(42,56)(44,50)(46,63)(47,59)(48,58)(49,60)(55,61)(57,64);(1,3,7,17)(2,8,13,34)(4,14,22,47)(5,24,27,15)(6,28,29,16)(9,35,38,57)(10,39,23,31)(11,32,12,33)(18,50,40,63)(19,44,21,46)(20,48,25,45)(26,49,56,55)(30,36,58,37)(41,61,42,60)(43,52,59,51)(53,62,54,64);(1,4)(2,9)(3,14)(5,25)(6,19)(7,22)(8,30)(10,40)(11,37)(12,36)(13,38)(15,48)(16,44)(17,47)(18,23)(20,27)(21,29)(24,45)(26,54)(28,46)(31,59)(32,35)(33,57)(34,58)(39,43)(41,51)(42,52)(49,62)(50,60)(53,56)(55,64)(61,63)

Polycyclic presentation:

2^9|b^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fgh,h^a=hi,c^b=cf,e^b=ei,g^b=gh,d^c=dg,e^c=ei,g^c=gi,h^c=hi,f^d=fhi,g^d=gi,g^f=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60318);