#147153

order	512
nilpotency class	3
isoclinism	isoclinic class of #453
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	#120
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1850
prime radical	C₂⁴	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

542 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:

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

Polycyclic presentation:

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

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, 54349);