#493472

order	512
nilpotency class	4
isoclinism	isoclinic class of #493464
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1782
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3428 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

2123 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:

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

Polycyclic presentation:

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

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, 400668);