#482710

order	512
nilpotency class	4
isoclinism	isoclinic class of #482709
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17595
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	#3060	C₂²
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #2875 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

669 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)(3,15,18,45)(4,21,23,35)(5,26,28,36)(6,13)(7,10)(8,14)(9,17)(11,24,38,25)(12,30,41,31)(16,46)(19,39,20,43)(22,40)(27,42)(29,37)(32,53)(33,56,63,60)(34,51)(44,49)(47,54,52,61)(48,62)(50,64)(55,58)(57,59);(1,3,8,18)(2,9,14,34)(4,16,23,50)(5,27,28,53)(6,20,7,19)(10,32,13,42)(11,33,38,63)(12,39,41,43)(15,36,45,26)(17,30,51,31)(21,54,35,61)(22,56,29,60)(24,49,25,48)(37,64,40,46)(44,55,62,59)(47,57,52,58);(2,10,14,13)(5,28)(9,32,34,42)(11,35,38,21)(12,26,41,36)(15,43,45,39)(17,51)(22,29)(27,53)(30,31)(33,61,63,54)(37,55,40,59)(44,64,62,46)(47,52)(56,60)(57,58);(1,4)(2,11,14,38)(3,16)(5,29)(6,24)(7,25)(8,23)(9,33,34,63)(10,35,13,21)(12,40,41,37)(15,44,45,62)(17,52)(18,50)(19,48)(20,49)(22,28)(26,59,36,55)(27,60)(30,58)(31,57)(32,61,42,54)(39,46,43,64)(47,51)(53,56)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,d^2=gh,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=ch,d^a=dg,e^a=eg,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,h^b=hi,e^c=ei,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240824, #72250033, #72250043, …
maximal normal subgroup of	#72240824, #72250033, #72250043, …
maximal quotient of	#72240808

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389906);