#105479

order	512
nilpotency class	3
isoclinism	isoclinic class of #8101
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	C₂³	C₄³
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄×#95
prime radical	C₂⁴	C₂×C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

600 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,f^2=i,g^2=h,b^a=bd,d^a=dh,f^a=fi,c^b=ci,d^b=di,e^b=eh

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, 12675);