#479488

order	512
nilpotency class	3
isoclinism	isoclinic class of #478580
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2789
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17210
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 555 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)(3,15,19,34,18,9,20,35)(4,21,24,10,26,38,25,39)(5,27,30,41,31,42,29,11)(16,45,48,43,47,36,22,32)(17,49,28,44,51,40,52,33)(23,55,57,63,58,37,56,53)(46,61,54,64,50,60,59,62);(1,3,6,19,8,18,7,20)(2,9,12,34,14,15,13,35)(4,22,24,16,26,48,25,47)(5,28,29,17,31,52,30,51)(10,36,21,43,39,45,38,32)(11,40,27,33,41,49,42,44)(23,50,56,54,58,46,57,59)(37,60,53,62,55,61,63,64);(1,4,6,24,8,26,7,25)(2,10,12,21,14,39,13,38)(3,16,20,22,18,47,19,48)(5,23,29,56,31,58,30,57)(9,32,35,45,15,43,34,36)(11,37,27,53,41,55,42,63)(17,50,28,59,51,46,52,54)(33,61,40,62,44,60,49,64);(1,5,6,29,8,31,7,30)(2,11,12,27,14,41,13,42)(3,17,20,28,18,51,19,52)(4,23,24,56,26,58,25,57)(9,33,35,40,15,44,34,49)(10,37,21,53,39,55,38,63)(16,46,22,54,47,50,48,59)(32,60,45,64,43,61,36,62)

Polycyclic presentation:

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

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, 386684);