#148917

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₈×D₄
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

490 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)(9,10,15,21,16,22,32,33,13,19,28,37,14,20,24,40)(11,23,27,35,26,38,31,39,25,17,12,18,30,34,29,36);(1,2,4,3)(5,7,8,6)(9,11)(10,17)(12,28)(13,25)(14,30)(15,27)(16,26)(18,21)(19,23)(20,38)(22,34)(24,29)(31,32)(33,36)(35,37)(39,40);(1,3,4,2)(6,7)(9,12,13,27)(10,18,19,35)(11,24,25,32)(14,31,16,29)(15,30,28,26)(17,33,23,40)(20,39,22,36)(21,34,37,38)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=cf,f^b=fi,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, 56113);