#146844

order	512
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1923
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

496 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2,3,4)(5,6,10,17,12,16,34,42,13,18,11,35,9,32,33,59)(7,19,22,44,24,39,52,60,25,45,23,53,21,14,36,38)(8,26,29,56,31,55,58,66,28,15,30,41,27,40,57,64)(20,46,49,67,51,37,62,63,48,43,50,65,47,54,61,68);(5,7,11,23,12,24,33,36,13,25,10,22,9,21,34,52)(6,14,17,38,16,19,42,44,18,39,35,60,32,45,59,53)(8,20,30,50,31,51,57,61,28,48,29,49,27,47,58,62)(15,37,41,63,40,43,64,65,26,54,56,68,55,46,66,67);(5,8,9,27,13,28,12,31)(6,15,16,40,18,26,32,55)(7,20,21,47,25,48,24,51)(10,29,33,57,11,30,34,58)(14,37,19,43,39,54,45,46)(17,41,42,64,35,56,59,66)(22,49,36,61,23,50,52,62)(38,63,44,65,60,68,53,67)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54040);