#545946

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2990 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2,4,3)(5,6,9,11)(7,8,12,10)(14,18)(16,20)(22,24)(26,27);(5,7,9,12)(6,10,11,8)(13,14)(15,16)(17,18)(19,20)(21,22,25,24)(23,27,28,26);(1,3,4,2)(6,11)(7,12)(14,18)(15,19)(21,22,25,24)(23,27,28,26);(5,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,19)(18,20)(21,23,25,28)(22,26,24,27);(5,8)(6,7)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,24,25,22)(23,27,28,26)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 453142);