#151141

order	512
nilpotency class	5
isoclinism	isoclinic class of #151138
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12090
commutator, Frattini, cosocle	#407	C₂³
prime radical	#23899	C₂
LCS	G ▷ #407 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #407 ◁ G
LECS	G ▷ #407 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

566 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 38 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=fi,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=df,e^a=ei,f^a=fi,g^a=gh,d^b=df,e^b=egh,f^b=fi,g^b=gi,h^b=hi,d^c=dg,e^c=ei,f^c=fi,e^d=ehi,g^d=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, 58337);