#542563

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

3192 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fghi,c^2=f,d^2=f,e^2=gi,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, 449759);