#6466289

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1770 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373485);