#152646

order	512
nilpotency class	5
isoclinism	isoclinic class of #152644
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13059
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	#33611	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

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

normal subgroups only

This group has 306 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,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,e^b=ei,d^c=di,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224990, #72224995, #72225001, …
maximal normal subgroup of	#72224990, #72224995, #72225001, …
maximal quotient of	C₂×#152646

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59842);