#472528

order	512
nilpotency class	3
isoclinism	isoclinic class of #456381
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2712
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16823
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

844 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)(3,14)(4,9)(5,25,28,10)(6,12)(7,13)(8,17)(11,30)(15,43)(16,44,49,62)(18,33)(19,41)(20,31)(21,53,54,35)(22,42)(23,36)(24,37)(26,38,55,57)(27,39,29,40)(32,48,56,50)(34,46)(45,63,58,61)(47,64,52,59)(51,60);(1,3,7,19)(2,8,13,33)(4,20)(5,16,28,49)(6,18,30,17)(9,34)(10,32,25,56)(11,41,12,14)(15,22)(21,52)(23,46)(24,51)(26,58)(27,50,29,48)(31,42)(35,61)(36,60)(37,43)(38,59)(39,44,40,62)(45,55)(47,54)(53,63)(57,64);(1,4)(2,9)(3,15)(5,26,28,55)(6,22)(7,24)(8,31)(10,38,25,57)(11,36)(12,42)(13,37)(14,43)(16,47,49,52)(17,20)(18,51)(19,46)(21,27,54,29)(23,30)(32,61,56,63)(33,60)(34,41)(35,39,53,40)(44,59,62,64)(45,48,58,50);(1,5)(2,10)(3,16)(4,21)(6,29)(7,28)(8,32)(9,35)(11,40)(12,39)(13,25)(14,44)(15,45)(17,50)(18,48)(19,49)(20,52)(22,55)(23,26)(24,54)(27,30)(31,59)(33,56)(34,61)(36,57)(37,53)(38,42)(41,62)(43,63)(46,58)(47,51)(60,64)

Polycyclic presentation:

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

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, 379724);