#7529193

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1464 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)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,26)(21,35)(22,28)(23,27)(24,31)(29,41)(30,39)(32,48)(33,45)(34,44)(36,40)(37,43)(38,47)(42,46);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,32)(21,30)(22,31)(23,36)(25,28)(26,42)(27,39)(29,33)(34,46)(35,40)(37,47)(38,45)(41,43)(44,48);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,19)(18,25)(20,29)(21,36)(22,31)(23,30)(24,28)(26,41)(27,39)(32,33)(34,47)(35,40)(37,46)(38,44)(42,43)(45,48);(10,14)(13,16)(17,20,22,34)(18,26,28,44)(19,29,31,47)(21,33,23,37)(24,38,25,41)(27,43,35,45)(30,46,36,32)(39,42,40,48);(1,4,5,3)(2,7,8,6)(17,21,22,23)(18,27,28,35)(19,30,31,36)(20,33,34,37)(24,39,25,40)(26,43,44,45)(29,46,47,32)(38,42,41,48)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=fi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,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, 7436389);