#6467822

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	#24762	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1892 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)(7,10)(8,9)(11,12)(13,14)(15,25)(16,21)(17,22)(18,33)(19,24)(20,28)(23,30)(26,39)(27,34)(29,40)(31,37)(32,38)(35,42)(36,41)(43,44);(2,3)(13,15,18,28,19,29,30,41)(14,20,23,25,24,36,33,40)(16,26,17,27,31,42,32,43)(21,34,38,39,37,44,22,35);(1,2,4,3)(6,9)(10,12)(13,16)(14,21)(15,26)(17,18)(19,31)(20,34)(22,33)(23,38)(24,37)(25,39)(27,28)(29,42)(30,32)(35,40)(36,44)(41,43);(1,3,4,2)(13,17)(14,22)(15,27)(16,30)(18,31)(19,32)(20,35)(21,23)(24,38)(25,34)(26,41)(28,42)(29,43)(33,37)(36,39)(40,44);(5,7,8,11)(6,10,9,12)(16,31)(17,32)(21,37)(22,38)(26,42)(27,43)(34,44)(35,39)

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,d^c=di,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, 6375018);