#453794

order	512
nilpotency class	3
isoclinism	isoclinic class of #428398
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2609
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16116
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 581 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=f,c^2=e,d^2=eg,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,f^c=fi,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, 360990);