#7588944

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 ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1250 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=fi,d^2=fg,e^2=fg,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,e^d=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, 7496140);