#582529

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
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	C₂×#2656	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3286 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)(7,12)(8,9)(10,11)(14,17)(18,20)(21,22,24,26)(23,27,28,25);(1,3,4,2)(5,7)(6,9)(8,10)(11,12)(13,14)(15,18)(16,17)(19,20)(21,23,24,28)(22,25,26,27);(6,11)(7,8)(15,19)(18,20)(21,23,24,28)(22,25,26,27);(5,8,10,7)(6,9,11,12)(13,15)(14,18)(16,19)(17,20)(21,23)(22,25)(24,28)(26,27);(5,9)(6,7)(8,11)(10,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,25)(24,26)(27,28)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=h,b^a=bf,c^a=cg,d^a=dgh,e^a=ef,c^b=ch,d^b=dg,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, 489725);