#479891

order	512
nilpotency class	3
isoclinism	isoclinic class of #479233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2785
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17308
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

549 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 549 conjugacy classes of subgroups — too many to draw. Only its 152 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=efh,c^2=g,d^2=efg,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,g^b=gi,h^b=hi,d^c=de,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387087);