#7536280

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1649 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)(17,18)(19,25)(20,26)(21,27)(22,37)(23,28)(24,29)(30,39)(31,40)(32,42)(33,43)(34,45)(35,46)(36,44)(38,41)(47,48);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,24,33)(18,25,29,43)(20,30,21,35)(22,32,23,38)(26,39,27,46)(28,41,37,42)(31,34,47,36)(40,45,48,44);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20)(18,26)(19,30)(21,24)(22,36)(23,34)(25,39)(27,29)(28,45)(31,32)(33,35)(37,44)(38,47)(40,42)(41,48)(43,46);(10,14)(13,15)(17,21,24,20)(18,27,29,26)(19,31,33,47)(22,36,23,34)(25,40,43,48)(28,45,37,44)(30,38,35,32)(39,41,46,42);(2,5)(6,8)(10,14)(13,15)(17,22,24,23)(18,28,29,37)(19,32,33,38)(20,34,21,36)(25,41,43,42)(26,44,27,45)(30,47,35,31)(39,40,46,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fg,d^2=i,e^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,d^c=di,e^c=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7443476);