#495719

order	512
nilpotency class	4
isoclinism	isoclinic class of #2192
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₄×#447
prime radical	#33538	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1011 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402915);