#109567

order	512
nilpotency class	3
isoclinism	isoclinic class of #8304
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

600 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)(13,14,17,21,18,22,29,35)(15,23,26,37,25,19,30,32)(16,20,27,33,28,34,40,43)(24,36,39,44,38,31,41,42);(5,6,7,10,8,9,11,12)(13,15)(14,19,22,23)(16,24)(17,26)(18,25)(20,31,34,36)(21,32,35,37)(27,39)(28,38)(29,30)(33,42,43,44)(40,41);(2,4)(6,9)(10,12)(13,16,17,27,18,28,29,40)(14,20,21,33,22,34,35,43)(15,24,26,39,25,38,30,41)(19,31,32,42,23,36,37,44)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,f^2=h,g^2=i,b^a=bd,c^a=ce,g^a=gh,c^b=ci,d^b=dh

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, 16763);