#495894

order	512
nilpotency class	4
isoclinism	isoclinic class of #33618
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#466
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₄×#466
prime radical	#33617	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

633 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217287, #72217290, #72224949, …
maximal normal subgroup of	#72217287, #72217290, #72224949, …
maximal quotient of	#72225539, C₂×#495894

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403090);