#495804

order	512
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	#121
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₈×#121
prime radical	#3067	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

637 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320529, #72320727, #72320755
maximal normal subgroup of	#72320529, #72320727, #72320755

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403000);