#150857

order	512
nilpotency class	5
isoclinism	isoclinic class of #150854
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11880
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	#423	C₂³
prime radical	#24287	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

403 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 403 conjugacy classes of subgroups — too many to draw. Only its 60 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)(3,13,16,35,18,41,17,49)(4,19)(5,10,26,37)(6,28)(7,11)(8,15,30,47,31,48,39,23)(9,21)(12,25)(14,42,44,61,46,63,45,40)(20,51,24,57)(22,52)(27,60,54,38)(29,53,50,43,32,58,62,64)(33,59,36,56)(34,55);(1,3)(2,8)(4,14)(5,23)(6,16)(7,18)(9,32)(10,35)(11,31)(12,39)(13,38)(15,27)(17,25)(19,50)(20,53)(21,44)(22,46)(24,58)(26,47)(28,30)(29,34)(33,63)(36,42)(37,49)(40,57)(41,60)(43,56)(45,55)(48,54)(51,61)(52,62)(59,64);(1,4,6,21,7,22,25,55)(2,9,12,19,11,34,28,52)(3,14,16,44,18,46,17,45)(5,24,27,59,26,20,54,56)(8,29,39,62,31,32,30,50)(10,36,38,57,37,33,60,51)(13,40,49,63,41,61,35,42)(15,43,47,58,48,64,23,53)

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58053);