#415594

order	512
nilpotency class	3
isoclinism	isoclinic class of #403928
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2615
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15202
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

911 conjugacy classes of subgroups; only the 260 normal subgroups are shown.

normal subgroups only

This group has 911 conjugacy classes of subgroups — too many to draw. Only its 260 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=g,c^2=g,d^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322790);