#564580

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2442 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=f,d^2=fi,e^2=h,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^b=ef,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, 471776);