#149333

order	512
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₂×#102
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

524 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=hi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,f^b=fi,h^b=hi,d^c=dh,e^c=ehi,f^c=fi,h^c=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56529);