#6380183

order	512
nilpotency class	3
isoclinism	isoclinic class of #61106
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×D₄²
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×D₄×C₄⋊C₄
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2411 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285627, #72285848, #72285921
maximal normal subgroup of	#72285627, #72285848, #72285921
maximal quotient of	#72286428, #72342648

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287379);