#354453

order	512
nilpotency class	3
isoclinism	isoclinic class of #3019
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1586 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)(9,10,14,17,15,11,13,19)(12,16,20,24,21,18,22,23);(5,6,7,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,21)(20,24)(22,23);(9,12,15,21)(10,16,11,18)(13,20,14,22)(17,23,19,24);(1,3,4,2)(9,13,15,14)(10,17,11,19)(12,20,21,22)(16,23,18,24)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=eg,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,e^b=ei,f^b=fi,g^b=gi,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, 261649);