#6458219

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442595
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2487
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1749 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,g^2=i,b^a=bf,c^a=cg,g^a=gi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249825, #72249829, #72249831, …
maximal normal subgroup of	#72249825, #72249829, #72249831, …
maximal quotient of	#72257091, #72322821

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365415);