#7516530

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514172
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61075	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1925 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,g^c=gi,h^c=hi,e^d=ei,g^d=gi

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, 7423726);