#203513

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

806 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=chi

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, 110709);