#7148451

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1771 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)(7,11)(10,12)(13,14)(15,23)(16,20)(17,30)(18,21)(19,26)(22,31)(24,37)(25,34)(27,33)(28,36)(29,42)(32,40)(35,41)(38,44)(39,43);(6,9)(7,11)(13,15,18,27)(14,19,21,34)(16,24,28,39)(17,25,31,26)(20,32,36,44)(22,33,30,23)(29,38,41,40)(35,43,42,37);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,28)(14,20,21,36)(15,24,27,39)(17,29,31,41)(19,32,34,44)(22,35,30,42)(23,37,33,43)(25,38,26,40);(6,9)(10,12)(15,25)(17,31)(19,33)(22,30)(23,34)(24,38)(26,27)(29,41)(32,43)(35,42)(37,44)(39,40);(5,7,8,11)(6,10,9,12)(14,21)(16,28)(19,34)(22,30)(23,33)(24,39)(29,41)(38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7055647);