#105324

order	512
nilpotency class	3
isoclinism	isoclinic class of #7943
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#95
commutator	C₂×C₄	C₄³
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄×#95
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,4,9,6,10,5,8)(3,11,13,7,12,16,14,15)(17,18,21,27,23,26,22,28)(19,29,31,40,33,24,32,39)(20,25,35,43,37,42,36,44)(30,41,45,48,34,38,46,47);(1,3,5,14,6,12,4,13)(2,7,9,16,10,15,8,11)(17,19,21,32,23,33,22,31)(18,24,27,40,26,29,28,39)(20,34,35,45,37,30,36,46)(25,41,43,47,42,38,44,48);(3,12)(7,15)(11,16)(13,14)(17,20,22,36,23,37,21,35)(18,25,28,44,26,42,27,43)(19,30,31,45,33,34,32,46)(24,38,39,47,29,41,40,48)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=hi,e^2=h,f^2=h,g^2=i,b^a=bd,d^a=dh,f^a=fhi,c^b=ch,e^b=ei

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, 12520);