#561696

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, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=h,d^2=fg,e^2=f,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, 468892);