#353274

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂³×C₄²

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

1450 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)(17,18,20,24,22,26,21,25)(19,23,28,31,27,30,29,32);(9,10)(11,14)(12,13)(15,16)(17,19,21,29,22,27,20,28)(18,23,25,32,26,30,24,31);(5,6,7,8)(10,13)(14,16)(19,27)(23,30)(28,29)(31,32);(9,11,12,15)(10,14,13,16)

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=cei

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, 260470);