#516683

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, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2608 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=fh,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, 423879);