#356725

order	512
nilpotency class	3
isoclinism	isoclinic class of #3288
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1308 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)(9,10,14,19,16,12,15,24)(11,20,18,21,22,13,23,17);(5,6,7,8)(9,11)(10,17)(12,21)(13,19)(14,23)(15,18)(16,22)(20,24);(1,2,4,3)(9,12,16,10)(11,21,22,17)(13,23,20,18)(14,24,15,19);(1,3,4,2)(9,13,14,17,16,20,15,21)(10,18,19,22,12,23,24,11)

Polycyclic presentation:

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

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, 263921);