#354457

order	512
nilpotency class	3
isoclinism	isoclinic class of #3026
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2040 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 261653);