#494651

order	512
nilpotency class	4
isoclinism	isoclinic class of #13068
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1893
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1893
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1263 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)(3,14)(4,21,25,35)(5,11)(6,27)(7,30)(8,13)(9,18)(10,24,37,22)(12,15)(16,46,49,63)(17,41)(19,42)(20,43)(23,36,28,38)(26,39)(29,57)(31,58,61,64)(32,51)(33,44)(34,53)(40,47,60,50)(45,62,56,55)(48,54,52,59);(1,3,15,44,7,19,29,53,8,20,26,51,5,17,6,18)(2,9,27,41,11,32,39,43,13,34,57,42,30,33,12,14)(4,16,45,64,24,50,23,48,25,49,56,58,22,47,28,52)(10,31,55,46,35,59,36,60,37,61,62,63,21,54,38,40);(1,4)(2,10)(3,16)(5,22)(6,28)(7,24)(8,25)(9,31)(11,35)(12,38)(13,37)(14,40)(15,45)(17,47)(18,52)(19,50)(20,49)(21,30)(23,29)(26,56)(27,55)(32,59)(33,54)(34,61)(36,39)(41,46)(42,63)(43,60)(44,64)(48,53)(51,58)(57,62);(1,5,8,7)(2,11,13,30)(3,17,20,19)(4,22,25,24)(6,26,29,15)(9,32,34,33)(10,35,37,21)(12,27,39,57)(14,41,43,42)(16,47,49,50)(18,51,53,44)(23,45,28,56)(31,59,61,54)(36,62,38,55)(40,46,60,63)(48,64,52,58)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285693, #72285725, #72285749, …
maximal normal subgroup of	#72285693, #72285725, #72285749, …
maximal quotient of	#72321231, #72322763, #72342557

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401847);