#428804

order	512
nilpotency class	3
isoclinism	isoclinic class of #428803
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15390
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

1018 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 336000);