#353241

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂²×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

1606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=ef,f^2=i,b^a=be,c^a=ci,d^a=di,c^b=ci,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, 260437);