#496723

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#21973
prime radical	C₂×#41	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

439 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224951, #72224958, #72224961, …
maximal normal subgroup of	#72224951, #72224958, #72224961, …
maximal quotient of	C₂×#496723

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403919);