#483016

order	512
nilpotency class	4
isoclinism	isoclinic class of #483015
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1530
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#97	C₂⁴
prime radical	#3512	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2875 ◁ G
LECS	G ▷ #97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

696 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,e^2=h,f^2=g,g^2=i,b^a=be,c^a=cg,e^a=eg,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,d^c=di,e^c=ei,f^c=fi,f^e=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, 390212);