#149760

order	512
nilpotency class	5
isoclinism	isoclinic class of #149759
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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=h,d^2=i,e^2=g,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,d^b=di,e^b=efi,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, 56956);