#587473

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2974 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=g,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, 494669);