#452729

order	512
nilpotency class	3
isoclinism	isoclinic class of #429607
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2595
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16061
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

880 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)(3,15)(4,20,23,39)(5,26)(6,12)(7,13)(8,14)(9,18)(10,24,38,25)(11,29)(16,44,47,36)(17,48)(19,35)(21,43,46,32)(22,37,53,61)(27,57)(28,42)(30,41)(31,34)(33,49)(40,50)(45,62,58,59)(51,55,56,54)(52,60,64,63);(1,3)(2,9,14,15)(4,21)(5,27,30,17)(6,19)(7,31)(8,18)(10,36,38,44)(11,40)(12,34,13,35)(16,24)(20,43,39,32)(22,52,53,64)(23,46)(25,47)(26,48)(28,50,29,49)(33,42)(37,60)(41,57)(45,55,58,54)(51,62)(56,59)(61,63);(1,4,6,25,8,23,7,24)(2,10,12,39,14,38,13,20)(3,16,31,46,18,47,19,21)(5,22,28,55,30,53,29,54)(9,32,35,36,15,43,34,44)(11,37,26,56,42,61,41,51)(17,45,50,64,27,58,49,52)(33,59,48,60,40,62,57,63);(1,5)(2,11)(3,17)(4,22)(6,28)(7,29)(8,30)(9,33)(10,37)(12,26)(13,41)(14,42)(15,40)(16,45)(18,27)(19,49)(20,51)(21,52)(23,53)(24,54)(25,55)(31,50)(32,59)(34,57)(35,48)(36,60)(38,61)(39,56)(43,62)(44,63)(46,64)(47,58)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 359925);