#369946

order	512
nilpotency class	3
isoclinism	isoclinic class of #23900
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1242 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fi,e^c=ei,f^d=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 277142);