#149758

order	512
nilpotency class	5
isoclinism	isoclinic class of #149757
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11501
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	#407	C₂³
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2921 ◁ G
LECS	G ▷ #407 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

351 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 56 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=h,e^2=g,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,e^b=ef,g^b=gi,d^c=di,e^c=ei,e^d=eh,f^e=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, 56954);