#387093

order	512
nilpotency class	3
isoclinism	isoclinic class of #213808
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1025 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294289);