#479872

order	512
nilpotency class	3
isoclinism	isoclinic class of #477321
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2830
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17295
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

571 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 571 conjugacy classes of subgroups — too many to draw. Only its 160 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)(3,15,19,35,20,34,18,9)(4,21,25,10,26,38,24,39)(5,27,30,42,31,41,29,11)(16,44,22,52,47,32,46,36)(17,33,49,43,51,60,50,59)(23,54,28,40,55,37,56,62)(45,61,53,57,64,58,48,63);(1,3,6,19,8,20,7,18)(2,9,13,15,14,35,12,34)(4,22,24,16,26,46,25,47)(5,17,30,49,31,51,29,50)(10,36,21,32,39,52,38,44)(11,33,27,43,42,60,41,59)(23,48,56,64,55,53,28,45)(37,61,40,63,54,58,62,57);(1,4,6,25,8,26,7,24)(2,10,12,21,14,39,13,38)(3,16,18,46,20,47,19,22)(5,28,30,55,31,56,29,23)(9,32,15,36,35,44,34,52)(11,40,41,54,42,62,27,37)(17,48,50,64,51,53,49,45)(33,58,43,63,60,61,59,57);(1,5,6,30,8,31,7,29)(2,11,12,41,14,42,13,27)(3,17,18,50,20,51,19,49)(4,23,25,28,26,55,24,56)(9,33,15,43,35,60,34,59)(10,37,21,40,39,54,38,62)(16,45,46,48,47,64,22,53)(32,57,36,58,44,63,52,61)

Polycyclic presentation:

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

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, 387068);