#391961

order	512
nilpotency class	3
isoclinism	isoclinic class of #391960
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14666
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

845 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 845 conjugacy classes of subgroups — too many to draw. Only its 192 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)(3,8,15,32)(4,17)(5,10)(6,25)(7,11)(9,20)(12,28)(13,42,18,51)(14,30,46,59)(16,49,48,31)(19,52)(21,40)(22,36,47,60)(23,41)(24,38)(26,35)(27,39)(29,50,33,44)(34,56)(37,54)(43,62,45,64)(53,61,57,58)(55,63);(1,3)(2,8,11,32)(4,18,21,13)(5,22)(6,16)(7,15)(9,33)(10,36,38,60)(12,31,25,49)(14,23)(17,51)(19,53,55,57)(20,50,26,44)(24,47)(27,46)(28,48)(29,35)(30,39,59,41)(34,62)(37,64)(40,42)(43,54,45,56)(52,61)(58,63);(1,4)(2,9)(3,13)(5,19)(6,26)(7,21)(8,29)(10,37)(11,35)(12,40)(14,45)(15,18)(16,50)(17,25)(20,28)(22,57)(23,56)(24,55)(27,54)(30,58)(31,51)(32,33)(34,38)(36,62)(39,63)(41,52)(42,49)(43,46)(44,48)(47,53)(59,61)(60,64);(1,5,7,24)(2,10,11,38)(3,14,15,46)(4,19,21,55)(6,27,28,23)(8,30,32,59)(9,34,35,37)(12,41,25,39)(13,43,18,45)(16,22,48,47)(17,52,40,63)(20,56,26,54)(29,58,33,61)(31,60,49,36)(42,62,51,64)(44,57,50,53)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299157);