#152171

order	512
nilpotency class	5
isoclinism	isoclinic class of #152169
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12731
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#3415	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

634 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 104 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224991, #72224995, #72224996, …
maximal normal subgroup of	#72224991, #72224995, #72224996, …
maximal quotient of	C₂×#152171

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59367);