#561003

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

2102 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=fg,e^2=gh,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, 468199);