#7143786

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45134	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2237 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,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, 7050982);