#493541

order	512
nilpotency class	4
isoclinism	isoclinic class of #12741
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1786
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1786
prime radical	C₂×#3584	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3125 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307095, #72307097, #72307103, …
maximal normal subgroup of	#72307095, #72307097, #72307103, …
maximal quotient of	C₂×#493541

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400737);