#406829

order	512
nilpotency class	3
isoclinism	isoclinic class of #393988
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1206 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 314025);