#6466074

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2020 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)(15,25,29,36)(16,30,17,22)(18,23,19,33)(21,34,37,28)(26,42,27,41)(31,40,32,39)(35,44,38,43);(2,3)(13,15,20,29)(14,21,24,37)(16,27,17,26)(18,28,19,34)(22,38,30,35)(23,36,33,25)(31,44,32,43)(39,42,40,41);(1,2,4,3)(5,7,8,12)(6,9,10,11)(13,16,20,17)(14,22,24,30)(15,26,29,27)(18,32,19,31)(21,35,37,38)(23,40,33,39)(25,41,36,42)(28,44,34,43);(1,3,4,2)(13,17,20,16)(14,22,24,30)(15,27,29,26)(18,31,19,32)(21,35,37,38)(23,40,33,39)(25,41,36,42)(28,43,34,44);(13,18,20,19)(14,23,24,33)(15,28,29,34)(16,31,17,32)(21,36,37,25)(22,39,30,40)(26,43,27,44)(35,41,38,42)

Polycyclic presentation:

2^9|a^2=g,b^2=i,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,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, 6373270);