#7566021

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	C₂×#3269	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1988 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=fgi,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, 7473217);