#483014

order	512
nilpotency class	4
isoclinism	isoclinic class of #11912
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1530
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#97	C₂⁴
socle	C₂	C₂×#1530
prime radical	#60682	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ #97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

706 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=i,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,e^c=ei,f^c=fi,f^e=fi

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, 390210);