#7536023

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1608 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(18,23)(24,36)(25,37)(26,34)(30,40)(38,47)(39,45)(41,48);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,18,21,23)(19,24,28,36)(20,30,31,40)(22,34,33,26)(25,35,37,32)(27,41,42,48)(29,45,44,39)(38,46,47,43);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,19,21,28)(18,24,23,36)(20,27,31,42)(22,29,33,44)(25,38,37,47)(26,39,34,45)(30,41,40,48)(32,43,35,46);(10,14)(13,15)(17,20,21,31)(18,25,23,37)(19,27,28,42)(22,35,33,32)(24,38,36,47)(26,40,34,30)(29,46,44,43)(39,48,45,41);(2,5)(6,8)(10,14)(13,15)(19,28)(24,36)(27,42)(29,44)(38,47)(39,45)(41,48)(43,46)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fg,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,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, 7443219);