#516565

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4454 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=fh,e^2=fh,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^c=df

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, 423761);