#413758

order	512
nilpotency class	3
isoclinism	isoclinic class of #413183
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2867
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15155
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1129 conjugacy classes of subgroups; only the 280 normal subgroups are shown.

normal subgroups only

This group has 1129 conjugacy classes of subgroups — too many to draw. Only its 280 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:

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

Polycyclic presentation:

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

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, 320954);