#6577469

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	C₂×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1992 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)(7,11)(8,10)(9,12)(14,19)(20,34)(21,35)(22,32)(23,38)(27,37)(36,43)(39,44);(5,7,9,8)(6,10,12,11)(13,14,17,19)(15,23,26,38)(16,27,29,37)(18,32,31,22)(20,33,34,24)(21,28,35,30)(25,36,41,43)(39,40,44,42);(2,4)(5,7,9,8)(6,11,12,10)(13,15,17,26)(14,20,19,34)(16,25,29,41)(18,33,31,24)(21,36,35,43)(22,38,32,23)(27,39,37,44)(28,42,30,40);(5,8,9,7)(6,10,12,11)(15,24)(16,28)(18,31)(20,23)(21,37)(22,32)(25,41)(26,33)(27,35)(29,30)(34,38)(36,43);(13,16,17,29)(14,21,19,35)(15,25,26,41)(18,28,31,30)(20,36,34,43)(22,37,32,27)(23,39,38,44)(24,40,33,42)

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=ehi,h^d=hi,h^e=hi

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, 6484665);