#6371292

order	512
nilpotency class	3
isoclinism	isoclinic class of #60388
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#120
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂²×#385
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4425 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317429, #72317446
maximal normal subgroup of	#72317429, #72317446

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278488);