#391214

order	512
nilpotency class	3
isoclinism	isoclinic class of #391213
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14621
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1064 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318430, #72318890, #72319003
maximal normal subgroup of	#72318430, #72318890, #72319003

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298410);