#493545

order	512
nilpotency class	4
isoclinism	isoclinic class of #481288
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#418
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1786
prime radical	C₂×#3584	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3127 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280507, #72280557, #72280969, …
maximal normal subgroup of	#72280507, #72280557, #72280969, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400741);