#152644

order	512
nilpotency class	5
isoclinism	isoclinic class of #152644
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	#33607	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

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

normal subgroups only

This group has 318 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)(3,15)(4,10,23,37)(5,26)(6,14)(7,33)(8,34)(9,18)(11,16)(12,28)(13,32)(17,46,47,41)(19,45)(20,44)(21,42)(22,53,27,61)(24,59,57,63)(25,60,58,39)(29,50)(30,52)(31,51)(35,49,54,38)(36,43,40,55)(48,62,64,56);(1,3,16,13,7,19,45,33,32,11,15,2,9,26,29,12,6,21,44,34,31,52,30,51,8,20,42,14,28,50,5,18)(4,17,43,39,24,48,56,59,58,40,41,10,35,61,22,38,23,47,55,60,57,64,62,63,25,36,46,37,54,53,27,49);(1,4)(2,10)(3,17)(5,27)(6,23)(7,24)(8,25)(9,35)(11,40)(12,38)(13,39)(14,37)(15,41)(16,43)(18,49)(19,48)(20,36)(21,47)(22,29)(26,61)(28,54)(30,62)(31,57)(32,58)(33,59)(34,60)(42,46)(44,55)(45,56)(50,53)(51,63)(52,64)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,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,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, #72224993, #72225003, …
maximal normal subgroup of	#72224990, #72224993, #72225003, …
maximal quotient of	C₂×#152644

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59840);