#153025

order	512
nilpotency class	5
isoclinism	isoclinic class of #153025
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13283
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#509
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

210 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 92 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)(3,16,21,50)(4,24,28,41)(5,30)(6,34)(7,38)(8,15)(9,19,29,58)(10,27,40,48)(11,17)(12,31)(13,36)(14,35)(18,33)(20,59,22,51)(23,62,60,26)(25,32)(37,43)(39,63,46,42)(44,47)(45,54)(49,57,55,61)(52,64)(53,56);(1,3,17,40,37,20,32,41,35,60,47,42,36,61,5,19,8,21,54,10,6,22,52,24,7,23,53,63,31,57,33,58)(2,9,18,55,12,46,56,62,38,28,64,51,34,27,45,50,15,29,30,49,13,39,44,26,14,4,25,59,43,48,11,16);(1,4,6,27,35,9,31,39,8,28,37,48,7,29,36,46)(2,10,12,24,38,63,34,58,15,40,13,41,14,42,43,19)(3,18,20,56,60,64,61,45,21,30,22,44,23,25,57,11)(5,26,17,51,52,16,47,49,33,62,54,59,32,50,53,55)

Polycyclic presentation:

2^9|a^2=f,b^2=def,c^2=e,d^2=ghi,e^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gh,h^a=hi,c^b=ch,e^b=ei,f^b=fi,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224990, #72224998, #72224999, …
maximal normal subgroup of	#72224990, #72224998, #72224999, …
maximal quotient of	C₂×#153025

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60221);