#7574884

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1362 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7482080);