#6483372

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470395
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2088 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6390568);