#536715

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

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

3182 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fghi,d^2=g,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, 443911);