#7552318

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

1356 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,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, 7459514);