#126648

order	512
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	D₈
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	C₂×#107
prime radical	C₂⁴	#107
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

510 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=gi,e^2=h,f^2=g,g^2=i,b^a=bd,d^a=dg,f^a=fgi,g^a=gi

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33844);