#107523

order	512
nilpotency class	3
isoclinism	isoclinic class of #8510
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

862 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi,e^c=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14719);