#391841

order	512
nilpotency class	3
isoclinism	isoclinic class of #213744
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1221 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299037);