#485593

order	512
nilpotency class	4
isoclinism	isoclinic class of #485592
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#18152
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	#2017	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

649 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 649 conjugacy classes of subgroups — too many to draw. Only its 116 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=i,d^2=h,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,g^a=gi,h^a=hi,e^b=eg,f^b=fhi,g^b=gi,d^c=di,e^c=eh,f^c=fi,e^d=ei,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, 392789);