#503268

order	512
nilpotency class	4
isoclinism	isoclinic class of #503268
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25779
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1072 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1072 conjugacy classes of subgroups — too many to draw. Only its 176 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)(3,13)(4,20)(5,24)(6,27)(7,11)(8,17)(9,21)(10,14)(12,25)(15,40)(16,43)(18,35)(19,48)(22,54)(23,50)(26,41)(28,55)(29,57)(30,53)(31,58)(32,39)(33,44)(34,37)(36,47)(38,49)(42,46)(45,56)(51,52)(59,62)(60,64)(61,63);(1,3,14,37,6,19,26,40,7,18,39,57,25,47,5,17)(2,8,24,36,12,29,32,35,11,15,41,48,27,34,10,13)(4,16,38,61,22,46,53,62,23,45,58,64,52,55,21,44)(9,28,51,60,31,56,50,59,30,42,54,63,49,43,20,33);(1,2)(3,15,18,8)(4,9)(5,10)(6,12)(7,11)(13,17,35,40)(14,24)(16,42,45,28)(19,34,47,29)(20,21)(22,31)(23,30)(25,27)(26,32)(33,44,59,62)(36,37,48,57)(38,51)(39,41)(43,55,56,46)(49,52)(50,53)(54,58)(60,61,63,64);(1,4,14,38,6,22,26,53,7,23,39,58,25,52,5,21)(2,9,24,51,12,31,32,50,11,30,41,54,27,49,10,20)(3,16,37,61,19,46,40,62,18,45,57,64,47,55,17,44)(8,28,36,60,29,56,35,59,15,42,48,63,34,43,13,33)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320162, #72343953, #72343961
maximal normal subgroup of	#72320162, #72343953, #72343961

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410464);