#435816

order	512
nilpotency class	3
isoclinism	isoclinic class of #435815
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15538
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×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)(3,13)(4,9)(5,21,24,10)(6,25)(7,12)(8,16)(11,28)(14,40)(15,35,44,50)(17,37)(18,46,49,32)(19,38)(20,34)(22,30,53,58)(23,36,55,51)(26,31)(27,33)(29,42)(39,43)(41,59,63,61)(45,57)(47,56,52,64)(48,60,54,62);(1,3)(2,8)(4,14)(5,22)(6,26)(7,17)(9,29)(10,35)(11,37)(12,31)(13,25)(15,23)(16,28)(18,47)(19,45)(20,43)(21,50)(24,53)(27,42)(30,36)(32,59)(33,39)(34,57)(38,40)(41,48)(44,55)(46,61)(49,52)(51,58)(54,63)(56,60)(62,64);(1,4,7,20)(2,9,12,34)(3,14,17,43)(5,18,24,49)(6,27,28,19)(8,29,31,57)(10,32,21,46)(11,38,25,33)(13,39,37,40)(15,41,44,63)(16,45,26,42)(22,52,53,47)(23,54,55,48)(30,56,58,64)(35,61,50,59)(36,62,51,60);(1,5)(2,10)(3,15,17,44)(4,18)(6,23)(7,24)(8,30,31,58)(9,32)(11,36)(12,21)(13,35,37,50)(14,41,43,63)(16,22,26,53)(19,48)(20,49)(25,51)(27,54)(28,55)(29,56,57,64)(33,60)(34,46)(38,62)(39,61,40,59)(42,47,45,52)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286106, #72286134, #72286235
maximal normal subgroup of	#72286106, #72286134, #72286235

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 343012);