#6491247

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3455	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2983 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)(13,14,20,34,19,25,18,24)(15,26,33,21,30,40,29,37)(16,22,32,39,31,38,17,23)(27,41,28,35,42,36,43,44);(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,27)(17,28)(18,33)(19,30)(20,29)(22,35)(23,36)(24,40)(25,37)(26,34)(31,42)(32,43)(38,44)(39,41);(2,4)(6,9)(10,12)(13,16)(14,22)(15,27)(17,18)(19,31)(20,32)(21,35)(23,24)(25,38)(26,41)(28,33)(29,43)(30,42)(34,39)(36,40)(37,44);(6,9)(10,12)(13,17,19,32)(14,23,25,39)(15,28,30,43)(16,18,31,20)(21,36,37,41)(22,24,38,34)(26,35,40,44)(27,33,42,29);(5,7,8,11)(6,10,9,12)(16,31)(17,32)(22,38)(23,39)(27,42)(28,43)(35,44)(36,41)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6398443);