#6450088

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
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

3048 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6357284);