#686983

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	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4478 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)(6,8)(13,14,18,19)(15,21,22,16)(17,20,25,26)(23,27,28,24);(6,8)(9,10,11,12)(14,19)(15,22)(17,25)(24,27);(5,6,7,8)(9,10)(11,12)(13,15,18,22)(14,16,19,21)(17,23,25,28)(20,24,26,27);(1,2)(3,4)(13,16)(14,15)(17,24)(18,21)(19,22)(20,23)(25,27)(26,28);(13,17)(14,20)(15,23)(16,24)(18,25)(19,26)(21,27)(22,28)

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=g,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, 594179);