#6451855

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

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 ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3912 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)(7,12)(8,10)(9,11)(13,14,19,22)(15,25,29,36)(16,21,31,40)(17,24,18,23)(20,28,35,27)(26,37,41,43)(30,34,42,44)(32,39,33,38);(2,4)(13,15,19,29)(14,20,22,35)(16,30,31,42)(17,28,18,27)(21,37,40,43)(23,36,24,25)(26,32,41,33)(34,38,44,39);(5,7,9,8)(6,10,11,12)(13,16)(14,21)(15,26)(17,33)(18,32)(19,31)(20,34)(22,40)(23,39)(24,38)(25,37)(27,30)(28,42)(29,41)(35,44)(36,43);(5,8,9,7)(6,10,11,12)(15,27)(17,18)(20,25)(23,24)(26,30)(28,29)(32,33)(34,37)(35,36)(38,39)(41,42)(43,44);(14,22)(16,31)(20,35)(23,24)(25,36)(26,41)(30,42)(32,33)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6359051);