#7517426

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45092	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1876 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=fi,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,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, 7424622);