#535457

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
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

2138 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 442653);