#686899

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

5166 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=f,d^2=f,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, 594095);