#109024

order	512
nilpotency class	3
isoclinism	isoclinic class of #1705
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#127
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#390
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

750 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,d^2=h,e^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210308, #72210667
maximal normal subgroup of	#72210308, #72210667
maximal quotient of	#72210504

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16220);