#470935

order	512
nilpotency class	3
isoclinism	isoclinic class of #455830
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2648
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16718
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

680 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)(4,21)(5,27,30,11)(6,13)(7,12)(8,14)(9,18)(10,24)(16,32)(17,43,50,60)(19,35)(20,34)(22,53)(23,40,55,52)(25,39)(26,38)(28,37,56,62)(29,41,31,42)(33,48,57,49)(36,45)(44,58,63,64)(46,51)(47,61,54,59);(1,3,8,20)(2,9)(4,22,26,45)(5,17,30,50)(6,18,7,19)(10,36)(11,33)(12,15)(13,34)(14,35)(16,24,46,25)(21,51)(23,54,55,47)(27,57)(28,44,56,63)(29,48,31,49)(32,38)(37,61)(39,53)(40,58)(41,60)(42,43)(52,64)(59,62);(1,4,6,24,8,26,7,25)(2,10,12,21,14,39,13,38)(3,16,19,45,20,46,18,22)(5,28,29,55,30,56,31,23)(9,32,34,53,35,51,15,36)(11,40,41,62,27,52,42,37)(17,47,49,44,50,54,48,63)(33,59,43,58,57,61,60,64);(1,5)(2,11)(3,17)(4,23)(6,31)(7,29)(8,30)(9,33)(10,37)(12,42)(13,41)(14,27)(15,43)(16,44)(18,49)(19,48)(20,50)(21,52)(22,54)(24,56)(25,28)(26,55)(32,58)(34,60)(35,57)(36,61)(38,40)(39,62)(45,47)(46,63)(51,64)(53,59)

Polycyclic presentation:

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

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, 378131);