#354360

order	512
nilpotency class	3
isoclinism	isoclinic class of #3068
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1182 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2,4,3)(9,10)(11,13)(12,20)(14,17)(15,18)(16,22)(19,23)(21,24);(5,6,7,8)(9,11,14,18,15,17,13,10)(12,19,23,20,22,24,21,16);(9,12,13,21,15,22,14,23)(10,16,17,24,18,20,11,19);(1,3,4,2)(9,13,15,14)(10,11,18,17)(12,21,22,23)(16,19,20,24)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=gi,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^b=di,f^c=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261556);