#147361

order	512
nilpotency class	3
isoclinism	isoclinic class of #453
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

740 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

(5,6,10,16,11,17,29,35,12,14,30,34,9,15,23,32)(7,18,20,28,19,27,13,31,21,8,22,25,26,24,33,36);(1,2,3,4)(5,7)(6,8)(9,26)(10,22)(11,19)(12,21)(13,23)(14,18)(15,27)(16,28)(17,24)(20,30)(25,34)(29,33)(31,35)(32,36);(5,8,23,36,9,27,30,25,12,18,29,31,11,24,10,28)(6,13,32,26,15,20,34,21,14,33,35,19,17,22,16,7)

Polycyclic presentation:

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

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, 54557);