#494459

order	512
nilpotency class	4
isoclinism	isoclinic class of #33531
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1874
prime radical	#60450	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1365 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307854, #72307924, #72308066, …
maximal normal subgroup of	#72307854, #72307924, #72308066, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401655);