#6369831

order	512
nilpotency class	3
isoclinism	isoclinic class of #2883
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂²×C₂²⋊C₄
commutator	C₂²	C₂³×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂²×C₄×C₂²⋊C₄
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4897 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,g^2=i,b^a=bf,h^a=hi,f^b=fi,e^d=ei

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317340, #72317342, #72317365, …
maximal normal subgroup of	#72317340, #72317342, #72317365, …
maximal quotient of	D₄×#1757

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277027);