#7548436

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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1404 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,13)(14,16)(17,18,23,27)(19,24,31,41)(20,25,33,44)(21,34,22,26)(28,37,46,48)(29,38,45,47)(30,40,36,39)(32,43,35,42);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,28)(21,30)(22,36)(23,31)(25,37)(26,39)(27,41)(29,32)(33,46)(34,40)(35,45)(38,43)(42,47)(44,48);(1,4,5,3)(2,6,7,8)(10,13)(14,15)(17,20,23,33)(18,25,27,44)(19,28,31,46)(21,35,22,32)(24,37,41,48)(26,43,34,42)(29,30,45,36)(38,40,47,39);(9,11,12,16)(10,14,13,15)(17,20)(18,25)(19,29)(21,32)(22,35)(23,33)(24,38)(26,42)(27,44)(28,36)(30,46)(31,45)(34,43)(37,39)(40,48)(41,47);(2,7)(6,8)(17,21,23,22)(18,26,27,34)(19,30,31,36)(20,32,33,35)(24,39,41,40)(25,42,44,43)(28,29,46,45)(37,47,48,38)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=g,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, 7455632);