#586010

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

2106 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)(5,6)(7,11)(8,9)(10,12)(13,14,16,18)(15,19,20,17)(21,22)(23,28)(24,27)(25,26);(6,8)(7,12)(13,14,16,18)(15,17,20,19)(21,23)(22,27)(24,26)(25,28);(5,7,9,12)(6,10,8,11)(13,14)(15,17)(16,18)(19,20)(21,22,26,25)(23,27,24,28);(5,7)(6,10)(8,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,24,26,23)(22,27,25,28);(5,8,9,6)(7,11,12,10)(13,15,16,20)(14,17,18,19)(21,25,26,22)(23,27,24,28)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=f,d^2=gh,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, 493206);