#500802

order	512
nilpotency class	4
isoclinism	isoclinic class of #500797
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24260
commutator, Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#3025	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1006 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312584, #72312588, #72312620, …
maximal normal subgroup of	#72312584, #72312588, #72312620

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407998);