#512840

order	512
nilpotency class	5
isoclinism	isoclinic class of #13667
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2184
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
socle	C₂	C₂×#2184
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

759 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 759 conjugacy classes of subgroups — too many to draw. Only its 144 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)(3,14)(4,9)(5,23)(6,27)(7,13)(8,16)(10,26)(11,21)(12,22)(15,42)(17,41)(18,48)(19,49)(20,50)(24,38)(25,55)(28,35)(29,34)(30,43)(31,45)(32,58)(33,53)(36,59)(37,62)(39,54)(40,46)(44,61)(47,63)(51,60)(52,56)(57,64);(1,3)(2,8)(4,15)(5,24)(6,28)(7,17)(9,30)(10,35)(11,38)(12,40)(13,31)(14,23)(16,26)(18,21)(19,22)(20,51)(25,57)(27,32)(29,44)(33,48)(34,60)(36,61)(37,63)(39,64)(41,55)(42,50)(43,53)(45,59)(46,62)(47,54)(49,56)(52,58);(5,25)(10,36)(14,41)(16,45)(20,52)(23,55)(24,57)(26,59)(33,37)(35,61)(42,49)(43,46)(48,63)(50,56)(51,58)(53,62);(1,4,7,22)(2,9,13,12)(3,15,17,19)(5,20,25,52)(6,21,29,54)(8,30,31,40)(10,33,36,37)(11,34,39,27)(14,42,41,49)(16,43,45,46)(18,44,47,28)(23,50,55,56)(24,51,57,58)(26,53,59,62)(32,38,60,64)(35,48,61,63)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224985, #72224994, #72225006
maximal normal subgroup of	#72224985, #72224994, #72225006

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420036);