#452456

order	512
nilpotency class	3
isoclinism	isoclinic class of #420424
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2699
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16047
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 823 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 359652);