#150563

order	512
nilpotency class	5
isoclinism	isoclinic class of #150560
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11796
commutator	C₂×C₂²⋊C₄	C₂²×C₄
Frattini, cosocle	#381	C₂³
prime radical	#3449	C₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ #2310 ◁ G
LECS	G ▷ #381 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

585 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 585 conjugacy classes of subgroups — too many to draw. Only its 54 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=g,e^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,d^b=dg,e^b=efg,h^b=hi,e^c=ei,e^d=eh,f^d=fi

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57759);