#370869

order	512
nilpotency class	3
isoclinism	isoclinic class of #23903
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

1088 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278065);