#7518271

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514148
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	#60433	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1881 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)(3,7)(4,5)(6,8)(9,10)(11,21)(12,24)(13,16)(14,19)(15,31)(17,28)(18,29)(20,30)(22,39)(23,34)(25,38)(26,35)(27,37)(32,36)(33,40);(2,5)(6,7)(9,11,14,13)(10,16,19,21)(12,25,27,40)(15,29,30,23)(17,35,36,39)(18,31,34,20)(22,32,26,28)(24,33,37,38);(9,12)(10,17)(11,22)(13,26)(14,27)(15,32)(16,33)(18,35)(19,36)(20,37)(21,38)(23,40)(24,31)(25,29)(28,30)(34,39);(9,13,14,11)(10,18,19,34)(12,26,27,22)(15,23,30,29)(16,20,21,31)(17,35,36,39)(24,33,37,38)(25,32,40,28);(1,3,4,8)(2,6,5,7)(10,19)(16,21)(17,36)(18,34)(20,31)(24,37)(33,38)(35,39)

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^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,g^d=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7425467);