#356343

order	512
nilpotency class	3
isoclinism	isoclinic class of #3290
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	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1090 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fi,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,d^b=de,e^b=ei,g^b=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, 263539);