#516477

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

2390 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fh,d^2=fg,e^2=g,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, 423673);