#586202

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

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

2172 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 493398);