#149365

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

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=deh,d^2=hi,e^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=dh,e^b=eh,g^b=gh,h^b=hi,d^c=dh,e^c=eh,g^c=gh,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, 56561);