#183580

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

868 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,e^2=h,f^2=h,g^2=i,b^a=be,d^a=dh,c^b=ci,e^b=eh,f^b=fh,d^c=dh

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, 90776);