#493917

order	512
nilpotency class	4
isoclinism	isoclinic class of #12873
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1823
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1823
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

989 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307092, #72307093, #72307096, …
maximal normal subgroup of	#72307092, #72307093, #72307096, …
maximal quotient of	C₂×#493917

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401113);