#499154

order	512
nilpotency class	4
isoclinism	isoclinic class of #499151
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2044
commutator, Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1077 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)(3,14)(4,20)(5,11)(6,27)(7,29)(8,13)(9,17)(10,24)(12,31)(15,43)(16,34)(18,49)(19,38)(21,53)(22,51)(23,56)(25,52)(26,59)(28,39)(30,42)(32,61)(33,47)(35,63)(36,55)(37,58)(40,48)(41,46)(44,50)(45,62)(54,64)(57,60);(1,3,8,19)(2,9,13,34)(4,21,25,15)(5,17,6,16)(7,30,31,18)(10,35,37,32)(11,38,27,14)(12,40,29,33)(20,50,52,62)(22,44,23,45)(24,54,58,46)(26,47,28,48)(36,41,60,64)(39,42,59,49)(43,56,53,51)(55,63,57,61);(1,4,6,23,8,25,5,22)(2,10,11,36,13,37,27,60)(3,15,17,45,19,21,16,44)(7,24,28,57,31,58,26,55)(9,32,14,41,34,35,38,64)(12,20,39,51,29,52,59,56)(18,46,48,61,30,54,47,63)(33,62,42,43,40,50,49,53);(1,5,8,6)(2,11,13,27)(3,16,19,17)(4,22,25,23)(7,26,31,28)(9,14,34,38)(10,36,37,60)(12,39,29,59)(15,44,21,45)(18,47,30,48)(20,51,52,56)(24,55,58,57)(32,41,35,64)(33,42,40,49)(43,50,53,62)(46,63,54,61)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303720, #72303740, #72303746, …
maximal normal subgroup of	#72303720, #72303740, #72303746, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406350);