#7003312

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#42908	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1902 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)(15,25)(16,22)(17,23)(18,34)(19,24)(20,28)(21,31)(26,42)(27,35)(29,37)(30,40)(32,38)(33,39)(36,44)(41,43);(5,6)(7,11)(8,10)(9,12)(13,15,17,32,19,29,18,28)(14,20,34,37,24,38,23,25)(16,26,33,27,31,44,30,43)(21,35,39,42,22,41,40,36);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,26)(17,33)(18,30)(19,31)(20,35)(22,24)(23,40)(25,36)(27,32)(28,43)(29,44)(34,39)(37,42)(38,41);(7,12)(9,11)(13,16)(14,22)(15,27)(17,30)(18,33)(19,31)(20,36)(21,24)(23,40)(25,35)(26,32)(28,44)(29,43)(34,39)(37,41)(38,42);(6,10)(9,11)(13,17,19,18)(14,23,24,34)(15,28,29,32)(16,30,31,33)(20,37,38,25)(21,39,22,40)(26,27,44,43)(35,36,41,42)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,e^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=ehi,d^c=dh,e^c=ei,f^d=fi

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, 6910508);