#152284

order	512
nilpotency class	5
isoclinism	isoclinic class of #152284
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12813
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

275 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 275 conjugacy classes of subgroups — too many to draw. Only its 88 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)(3,15,18,45)(4,11)(5,10)(6,28)(7,30)(8,14)(9,16,33,20)(12,24)(13,25)(17,43,23,56)(19,53,52,34)(21,35)(22,42)(26,38)(27,37)(29,59)(31,51,60,47)(32,61,40,63)(36,54)(39,55)(41,58)(44,57,62,49)(46,48,64,50);(1,3)(2,9,14,33)(4,16)(5,23)(6,19)(7,20)(8,18)(10,31,37,60)(11,15,30,45)(12,40,41,32)(13,34,42,53)(17,27)(21,51)(22,48)(24,57)(25,50)(26,47)(28,46,59,64)(29,52)(35,56,38,43)(36,44,39,62)(49,58)(54,61)(55,63);(2,10)(4,7)(5,24)(6,25)(9,31)(11,38)(12,13)(14,37)(15,43)(16,20)(17,49)(19,50)(21,54)(22,29)(23,57)(26,55)(27,58)(28,36)(30,35)(32,53)(33,60)(34,40)(39,59)(41,42)(44,46)(45,56)(47,63)(48,52)(51,61)(62,64)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224991, #72224997, #72224998, …
maximal normal subgroup of	#72224991, #72224997, #72224998, …
maximal quotient of	C₂×#152284

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59480);