#495805

order	512
nilpotency class	3
isoclinism	isoclinic class of #3068
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#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 ◁ C₈ ◁ C₂²×C₁₆ ◁ 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:

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

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,d^b=di,f^c=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224949, #72224961, #72224962, …
maximal normal subgroup of	#72224949, #72224961, #72224962, …
maximal quotient of	C₂×#495805

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403001);