#497150

order	512
nilpotency class	4
isoclinism	isoclinic class of #497148
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#22276
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33720	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1747 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 564 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)(3,16,12,30)(4,20)(5,28,10,18)(6,13)(7,33)(8,29)(9,15)(11,25)(14,34)(17,44,38,64)(19,36,41,31)(21,51,53,63)(22,54)(23,56,35,52)(24,48)(26,60)(27,50)(32,42)(37,47,57,45)(39,58)(40,55)(43,61,62,46)(49,59);(1,3,6,28,9,12,8,18)(2,10,13,30,15,5,29,16)(4,21,24,17,27,53,26,38)(7,19,32,61,22,41,34,46)(11,37,39,35,40,57,49,23)(14,36,33,62,42,31,54,43)(20,44,48,63,50,64,60,51)(25,52,58,45,55,56,59,47);(1,4,22,55,6,24,34,59,9,27,7,25,8,26,32,58)(2,11,14,20,29,49,54,60,15,40,42,50,13,39,33,48)(3,17,19,45,28,53,61,56,12,38,41,47,18,21,46,52)(5,23,36,63,30,57,43,44,10,35,31,51,16,37,62,64);(1,5,9,10)(2,12)(3,15)(4,23,27,35)(6,30,8,16)(7,31,22,36)(11,38)(13,28)(14,41)(17,40)(18,29)(19,42)(20,47)(21,49)(24,57,26,37)(25,51,55,63)(32,62,34,43)(33,61)(39,53)(44,58,64,59)(45,50)(46,54)(48,56)(52,60)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=efhi,d^2=ehi,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=dh,e^b=ei,g^b=gi,g^d=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404346);