#7003209

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	#42992	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1888 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)(7,9)(8,10)(11,12)(13,14)(15,24)(16,29)(17,21)(18,33)(19,22)(20,28)(23,34)(25,42)(26,35)(27,41)(30,36)(31,39)(32,40)(37,44)(38,43);(5,6)(7,11)(8,10)(9,12)(13,15,18,28,19,27,34,44)(14,20,23,24,22,37,33,41)(16,30,31,26,17,25,32,43)(21,38,39,42,29,35,40,36);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,25)(17,19)(18,31)(20,35)(22,29)(23,39)(24,36)(26,44)(27,30)(28,43)(32,34)(33,40)(37,38)(41,42);(7,12)(9,11)(13,17,19,16)(14,21,22,29)(15,26,27,43)(18,31,34,32)(20,36,37,42)(23,40,33,39)(24,35,41,38)(25,28,30,44);(6,10)(9,11)(14,22)(15,27)(16,17)(23,33)(28,44)(31,32)(35,38)(36,42)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=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, 6910405);