#687594

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#550	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

6856 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)(14,16)(19,23)(20,24)(26,27);(10,12)(13,14,18,16)(15,19,22,23)(17,24,25,20)(21,27,28,26);(5,6)(7,8)(9,10,11,12)(13,15)(14,19)(16,23)(17,21)(18,22)(20,26)(24,27)(25,28);(1,2)(3,4)(5,6)(7,8)(13,16,18,14)(15,19,22,23)(17,24,25,20)(21,26,28,27);(13,17,18,25)(14,20,16,24)(15,21,22,28)(19,26,23,27)

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=f,e^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, 594790);