#355030

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1002 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ef,d^2=eh,f^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262226);