#492881

order	512
nilpotency class	4
isoclinism	isoclinic class of #33388
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#102
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂²×#340
prime radical	C₂²×#41	D₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1561 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=hi,f^2=i,g^2=hi,h^2=i,b^a=be,e^a=eh,g^a=gh,h^a=hi,e^b=ei,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307047, #72307063, #72307128, …
maximal normal subgroup of	#72307047, #72307063, #72307128, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400077);