#369390

order	512
nilpotency class	3
isoclinism	isoclinic class of #24384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1454 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)(8,10)(9,12)(13,14,17,23,18,22,29,21)(15,19,25,35,26,34,41,33)(16,27,31,20,30,38,40,37)(24,39,42,32,28,43,44,36);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15,17,25,18,26,29,41)(14,19,23,35,22,34,21,33)(16,28,31,44,30,24,40,42)(20,36,38,39,37,32,27,43);(2,4)(5,7,9,8)(6,11,12,10)(13,16,29,40,18,30,17,31)(14,20,23,38,22,37,21,27)(15,24,41,44,26,28,25,42)(19,32,35,43,34,36,33,39);(5,8,9,7)(6,11,12,10)(16,30)(20,37)(24,28)(27,38)(31,40)(32,36)(39,43)(42,44)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=efg,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^c=di

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, 276586);