#148938

order	512
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₄×D₈
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

474 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(2,3)(5,6,8,7)(9,10,14,18,15,19,30,13)(11,16,21,32,23,33,22,28)(12,17,25,34,27,35,26,29)(20,31,36,39,37,40,38,24);(1,2,4,3)(5,7,8,6)(9,11)(10,16)(12,24)(13,28)(14,21)(15,23)(17,20)(18,32)(19,33)(22,30)(25,31)(26,40)(27,39)(29,38)(34,36)(35,37);(1,3,4,2)(6,7)(9,12)(10,17)(11,20)(13,29)(14,25)(15,27)(16,31)(18,34)(19,35)(21,36)(22,38)(23,37)(24,28)(26,30)(32,39)(33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,c^b=cf,f^b=fh,h^b=hi,f^c=fh,h^c=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56134);