#516824

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

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

3110 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=fg,d^2=fi,e^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, 424020);