#494059

order	512
nilpotency class	4
isoclinism	isoclinic class of #494058
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#20739
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#578	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2875 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

903 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)(3,9,16,33)(4,19)(5,11)(6,13)(7,27)(8,29)(10,22)(12,24)(14,40,42,63)(15,32,46,60)(17,48,47,62)(18,50,45,34)(20,35)(21,53)(23,55)(25,38)(26,52)(28,39)(30,37)(31,56,57,43)(36,54)(41,58,49,61)(44,64,51,59);(1,3)(2,9)(4,14)(5,15)(6,16)(7,17)(8,18)(10,31)(11,32)(12,34)(13,33)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)(35,58)(36,57)(37,59)(38,60)(39,61)(52,62)(53,63)(54,56)(55,64);(1,4,6,21)(2,10,13,36)(3,14,16,42)(5,22,25,54)(7,28,26,20)(8,30,24,23)(9,31,33,57)(11,37,38,55)(12,39,29,35)(15,43,46,56)(17,49,47,41)(18,51,45,44)(19,52,53,27)(32,59,60,64)(34,61,50,58)(40,62,63,48);(1,5,24,7,6,25,8,26)(2,11,12,27,13,38,29,52)(3,15,45,17,16,46,18,47)(4,20,30,54,21,28,23,22)(9,32,34,48,33,60,50,62)(10,19,35,37,36,53,39,55)(14,41,51,56,42,49,44,43)(31,40,58,59,57,63,61,64)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307113, #72307114, #72307115, …
maximal normal subgroup of	#72307113, #72307114, #72307115, …
maximal quotient of	C₂×#494059

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401255);