#6448725

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3280 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:

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

Polycyclic presentation:

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

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, 6355921);