#152169

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

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

normal subgroups only

This group has 636 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)(3,14,16,9)(4,8)(5,23)(6,26)(7,11)(10,19)(12,20)(13,24)(15,38,44,27)(17,48,46,54)(18,33,47,41)(21,37)(22,28)(25,55)(29,50)(30,49)(31,42,39,60)(32,43,40,61)(34,51)(35,53)(36,52)(45,57,62,59)(56,63,58,64);(2,8)(4,19)(5,20)(6,24)(9,27)(10,12)(11,28)(13,37)(14,38)(15,42)(17,43)(18,47)(21,53)(22,51)(23,29)(25,52)(26,30)(31,32)(33,59)(34,36)(35,55)(39,40)(41,57)(44,60)(45,64)(46,61)(48,58)(49,50)(54,56)(62,63);(1,3)(2,9)(4,15)(5,17)(6,18)(7,16)(8,27)(10,31)(11,14)(12,32)(13,33)(19,42)(20,43)(21,45)(22,44)(23,54)(24,47)(25,46)(26,41)(28,38)(29,56)(30,57)(34,39)(35,58)(36,40)(37,59)(48,55)(49,62)(50,63)(51,60)(52,61)(53,64)

Polycyclic presentation:

2^9|a^2=f,d^2=gh,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, #72224993, #72224994, …
maximal normal subgroup of	#72224991, #72224993, #72224994, …
maximal quotient of	C₂×#152169

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59365);