#108417

order	512
nilpotency class	3
isoclinism	isoclinic class of #105580
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	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

412 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)(3,7)(4,14,15,8)(5,10)(6,9)(11,23,24,18)(12,20)(13,19)(16,27,28,21)(17,22)(25,31,32,29)(26,30)(33,34)(35,43,46,56)(36,47,49,40)(37,41)(38,42)(39,45,52,60)(44,57)(48,62,63,53)(50,54)(51,58)(55,59)(61,64);(1,3,5,12,6,13,17,26)(2,7,10,20,9,19,22,30)(4,11,16,25,15,24,28,32)(8,18,21,29,14,23,27,31)(33,35)(34,39,42,52)(36,44)(37,45)(38,46)(40,51,47,61)(41,43,54,56)(48,58)(49,59)(50,60)(53,55,62,57)(63,64);(1,4)(2,8)(3,11)(5,16)(6,15)(7,18)(9,14)(10,21)(12,25)(13,24)(17,28)(19,23)(20,29)(22,27)(26,32)(30,31)(33,36,37,48,38,49,50,63)(34,40,41,53,42,47,54,62)(35,44,45,58,46,59,60,64)(39,51,43,55,52,61,56,57)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=i,g^2=hi,b^a=bd,c^a=ce,e^a=eh,g^a=gi,f^b=fi,f^c=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15613);