#396382

order	512
nilpotency class	3
isoclinism	isoclinic class of #396381
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14732
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1367 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303578);