#146933

order	512
nilpotency class	4
isoclinism	isoclinic class of #12833
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1807
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10820
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

469 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:

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

Polycyclic presentation:

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

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, 54129);