#512877

order	512
nilpotency class	5
isoclinism	isoclinic class of #512876
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2196
commutator, Frattini, cosocle	#124	C₂⁴
LCS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#124 ◁ G
LECS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1026 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 112 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)(4,9)(5,22)(6,12)(7,10)(8,15)(11,25)(14,45)(16,44)(17,51)(18,38)(19,54)(20,41)(21,34)(23,31)(24,39)(26,42)(27,36)(28,40)(29,37)(30,46)(32,50)(33,53)(35,56)(43,47)(48,63)(49,61)(52,64)(55,60)(57,58)(59,62);(1,3,7,18)(2,8,10,31)(4,14,21,48)(5,23,25,15)(6,17,29,16)(9,30,34,60)(11,38,22,13)(12,33,37,32)(19,55,56,46)(20,47,27,52)(24,59,57,49)(26,53,28,50)(35,63,54,45)(36,61,41,62)(39,64,58,43)(40,51,42,44);(1,4)(2,9)(3,14)(5,24)(6,27)(7,21)(8,30)(10,34)(11,39)(12,41)(13,43)(15,49)(16,47)(17,52)(18,48)(19,26)(20,29)(22,58)(23,59)(25,57)(28,56)(31,60)(32,61)(33,62)(35,40)(36,37)(38,64)(42,54)(44,45)(46,50)(51,63)(53,55);(2,10)(8,31)(9,34)(11,22)(12,37)(13,38)(30,60)(32,33)(35,54)(36,41)(39,58)(40,42)(43,64)(44,51)(45,63)(61,62)

Polycyclic presentation:

2^9|b^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,g^a=gh,e^b=ei,f^b=fghi,h^b=hi,e^c=eg,g^c=gi,h^c=hi,f^e=fhi,g^e=gi,g^f=gi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279927, #72286041
maximal normal subgroup of	#72279927, #72286041

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420073);