#150878

order	512
nilpotency class	5
isoclinism	isoclinic class of #150878
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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	#33611	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

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

normal subgroups only

This group has 253 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)(4,21,25,8)(5,27)(6,29)(7,14)(9,16)(10,22,42,58)(11,17)(12,24)(13,30)(18,51,23,35)(19,55)(20,38)(26,34,60,43)(28,36)(31,57,44,47)(32,50,33,53)(37,52,41,59)(39,54)(40,49)(45,48)(46,56)(61,63,62,64);(2,8)(3,16)(4,22)(5,24)(6,30)(9,35)(10,12)(11,42)(13,43)(14,21)(15,44)(17,28)(18,52)(19,54)(20,49)(23,59)(25,58)(26,53)(27,33)(29,34)(31,38)(32,36)(37,39)(40,51)(41,45)(46,61)(48,56)(50,60)(55,62)(63,64);(1,3,17,48,30,56,5,16,7,20,24,54,6,19,28,49)(2,9,36,46,29,45,12,15,14,40,27,55,13,39,11,38)(4,23,50,64,60,52,22,57,25,18,53,63,26,59,58,47)(8,31,42,37,43,62,33,51,21,44,10,41,34,61,32,35)

Polycyclic presentation:

2^9|a^2=e,c^2=gh,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, #72287275, …
maximal normal subgroup of	#72286130, #72287254, #72287275, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58074);