#465193

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1204 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

(5,6)(7,8)(10,13)(14,16)(17,18,22,19)(20,27,30,32)(21,26,28,24)(23,31,25,29);(1,2,3,4)(9,10,12,13)(11,14,15,16)(17,19,22,18)(20,23,30,25)(21,26,28,24)(27,31,32,29);(2,4)(11,15)(14,16)(17,20,28,31,22,30,21,29)(18,23,26,32,19,25,24,27);(6,8)(9,11,12,15)(10,14,13,16)(18,19)(23,25)(24,26)(27,32)

Polycyclic presentation:

2^9|a^2=i,b^2=eg,c^2=fi,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di,d^c=de

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 372389);