#462042

order	512
nilpotency class	3
isoclinism	isoclinic class of #455576
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2730
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16414
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

714 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 369238);