#687083

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2539	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3895 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=g,d^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,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, 594279);