#556626

order	512
nilpotency class	2
isoclinism	isoclinic class of #37436
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁵×C₄²

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

2238 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)(5,6,9,7)(8,10,12,11)(13,14,17,16)(15,18,19,20)(21,22,25,24)(23,26,28,27);(6,7)(8,12)(14,16)(18,20)(21,22,25,24)(23,27,28,26);(6,7)(10,11)(15,19)(18,20)(21,23,25,28)(22,26,24,27);(1,3,4,2)(5,7,9,6)(8,11,12,10)(13,15)(14,18)(16,20)(17,19)(21,24,25,22)(23,27,28,26);(5,8,9,12)(6,10,7,11)(13,16,17,14)(15,20,19,18)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=fgi,e^2=f,b^a=bf,c^a=cg,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, 463822);