#516617

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

5702 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)(5,6,7,8)(13,14,18,15)(16,19,24,21)(17,22,25,20)(23,27,28,26);(2,4)(9,10,11,12)(14,15)(19,21)(20,22)(26,27);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,26,28,27);(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27);(13,17)(14,20)(15,22)(16,23)(18,25)(19,26)(21,27)(24,28)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=f,d^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^c=df

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 423813);