#353281

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂²	C₂×C₄³
socle, prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1434 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=e,f^2=i,g^2=i,b^a=be,d^a=de,c^b=ce,d^c=di

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260477);