#6380655

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2070 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gi,d^2=fi,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,d^b=df,f^b=fi,h^b=hi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287851);