#7615137

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1500 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(20,30)(24,39)(27,32)(31,46)(34,40)(36,41)(42,47)(44,48);(2,6)(7,8)(11,16)(13,15)(17,18,22,26)(19,23,28,37)(20,31,30,46)(21,35,33,25)(24,40,39,34)(27,42,32,47)(29,45,43,38)(36,48,41,44);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,19,22,28)(18,23,26,37)(20,32,30,27)(21,29,33,43)(24,41,39,36)(25,38,35,45)(31,47,46,42)(34,48,40,44);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,20,33,40,22,30,21,34)(18,24,35,46,26,39,25,31)(19,27,43,48,28,32,29,44)(23,36,45,47,37,41,38,42);(2,6)(7,8)(19,28)(20,30)(23,37)(24,39)(29,43)(31,46)(34,40)(38,45)

Polycyclic presentation:

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

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, 7522333);