#518003

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2678 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=fg,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^d=ef

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 425199);