#108484

order	512
nilpotency class	3
isoclinism	isoclinic class of #106017
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1434
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

383 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=i,e^2=i,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eh,g^a=gi,f^b=fi,e^c=ei,f^c=fhi

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, 15680);