#6444109

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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×#3061	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2148 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)(7,11)(8,12)(9,10)(13,14)(15,25)(16,29)(17,20)(18,22)(19,23)(21,33)(24,34)(26,41)(27,38)(28,36)(30,44)(31,37)(32,39)(35,43)(40,42);(1,2)(3,4)(6,10)(11,12)(13,15,19,17)(14,20,23,25)(16,26,32,30)(18,28,34,27)(21,35,37,40)(22,38,24,36)(29,44,39,41)(31,43,33,42);(2,4)(5,7,9,8)(6,11,10,12)(13,16,18,33,19,32,34,31)(14,21,24,29,23,37,22,39)(15,26,28,42,17,30,27,43)(20,35,36,44,25,40,38,41);(2,4)(5,8,9,7)(6,12,10,11)(13,17,19,15)(14,20,23,25)(16,30,32,26)(18,27,34,28)(21,35,37,40)(22,38,24,36)(29,44,39,41)(31,42,33,43);(13,18,19,34)(14,22,23,24)(15,27,17,28)(16,31,32,33)(20,36,25,38)(21,29,37,39)(26,42,30,43)(35,41,40,44)

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,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, 6351305);