#382820

order	512
nilpotency class	3
isoclinism	isoclinic class of #213788
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1323 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,22)(13,16)(14,17)(18,24)(19,20)(21,23);(11,18)(12,13)(15,23)(16,22)(19,24)(20,21);(2,5)(6,8)(9,11,13,21,14,20,12,18)(10,15,16,23,17,19,22,24);(1,3,4,7)(2,6,5,8)(9,10)(11,19)(12,16)(13,22)(14,17)(15,20)(18,24)(21,23)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290016);