#396831

order	512
nilpotency class	3
isoclinism	isoclinic class of #396106
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2459
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14739
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

942 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)(4,20)(5,26)(6,30)(7,12)(8,13)(9,17)(10,24)(11,29)(15,31)(16,45)(18,33)(19,41)(21,34)(22,52)(23,37)(25,36)(27,38)(28,58)(32,47)(35,53)(39,43)(40,48)(42,62)(44,49)(46,56)(50,55)(51,61)(54,57)(59,64)(60,63);(1,3,6,17,8,19,7,18)(2,9,12,14,13,33,30,41)(4,21,24,15,25,43,23,44)(5,16,27,46,29,48,28,47)(10,34,20,49,37,39,36,31)(11,32,38,40,26,56,58,45)(22,51,54,42,55,63,53,64)(35,60,50,62,57,61,52,59);(1,4)(2,10)(3,15)(5,22)(6,23)(7,24)(8,25)(9,31)(11,35)(12,36)(13,37)(14,39)(16,42)(17,21)(18,43)(19,44)(20,30)(26,57)(27,53)(28,54)(29,55)(32,59)(33,49)(34,41)(38,52)(40,61)(45,60)(46,51)(47,63)(48,64)(50,58)(56,62);(1,5,6,27,8,29,7,28)(2,11,12,38,13,26,30,58)(3,16,17,46,19,48,18,47)(4,22,23,53,25,55,24,54)(9,32,14,40,33,56,41,45)(10,35,36,52,37,57,20,50)(15,42,21,51,44,64,43,63)(31,59,39,61,49,62,34,60)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 304027);