#147242

order	512
nilpotency class	4
isoclinism	isoclinic class of #12952
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1864
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10885
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

515 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 54438);