#150879

order	512
nilpotency class	5
isoclinism	isoclinic class of #150878
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11886
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂×M₃₂	C₂³
prime radical	#33607	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

263 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 263 conjugacy classes of subgroups — too many to draw. Only its 68 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)(4,21,24,8)(5,26)(6,29)(7,14)(9,16)(10,22,41,58)(11,27)(12,17)(13,30)(18,52,23,35)(19,55)(20,37)(25,34,60,43)(28,42)(31,57,44,47)(32,54,33,51)(36,53,40,59)(38,50)(39,49)(45,48)(46,56)(61,63,62,64);(2,8)(3,16)(4,22)(5,17)(6,30)(9,35)(10,12)(11,41)(13,43)(14,21)(15,44)(18,53)(19,50)(20,49)(23,59)(24,58)(25,51)(26,33)(27,28)(29,34)(31,37)(32,42)(36,38)(39,52)(40,45)(46,61)(48,56)(54,60)(55,62)(63,64);(1,3,17,50,30,56,28,49,7,20,27,48,6,19,5,16)(2,9,26,55,29,45,11,37,14,39,42,46,13,38,12,15)(4,23,51,63,60,53,58,47,24,18,54,64,25,59,22,57)(8,31,10,40,43,62,32,35,21,44,41,36,34,61,33,52)

Polycyclic presentation:

2^9|a^2=e,c^2=ghi,d^2=ghi,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=ghi,h^a=hi,c^b=cf,d^b=dg,e^b=efghi,f^b=fhi,g^b=ghi,h^b=hi,d^c=di,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286130, #72287254, #72287260, …
maximal normal subgroup of	#72286130, #72287254, #72287260, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58075);