#493470

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	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂×#519 ◁ 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,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, 400666);