#6469981

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469971
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4231 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)(5,6)(7,9)(8,10)(11,12)(13,14)(15,24)(16,21)(17,22)(18,31)(19,23)(20,26)(25,39)(27,34)(28,36)(29,37)(30,38)(32,35)(33,40)(41,44)(42,43);(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,25)(17,26)(18,32)(19,27)(21,33)(22,34)(23,35)(24,31)(28,40)(29,41)(30,42)(36,43)(37,39)(38,44);(2,4)(7,11)(9,12)(13,16)(14,21)(15,25)(17,28)(18,29)(19,30)(20,33)(22,36)(23,38)(24,39)(26,40)(27,42)(31,37)(32,41)(34,43)(35,44);(7,11)(9,12)(13,17,19,18)(14,22,23,31)(15,26,27,32)(16,28,30,29)(20,34,35,24)(21,36,38,37)(25,40,42,41)(33,43,44,39);(6,10)(9,12)(13,17,19,18)(14,22,23,31)(15,26,27,32)(16,29,30,28)(20,34,35,24)(21,37,38,36)(25,41,42,40)(33,39,44,43)

Polycyclic presentation:

2^9|d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=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, 6377177);