#110523

order	512
nilpotency class	3
isoclinism	isoclinic class of #9345
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

577 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,9)(3,10)(4,6,11,8)(7,12)(13,16)(14,15)(17,18,21,26,23,27,22,36)(19,28,31,42,30,24,32,38)(20,25,33,39,35,40,34,46)(29,41,44,48,43,37,45,47);(2,6,9,8)(7,15,16,10)(17,19,22,32,23,30,21,31)(18,24,26,38,27,28,36,42)(20,29,34,45,35,43,33,44)(25,37,39,47,40,41,46,48);(1,3,11,13,5,14,4,12)(2,7,8,10,9,16,6,15)(17,20,21,33,23,35,22,34)(18,25,26,39,27,40,36,46)(19,29,31,44,30,43,32,45)(24,37,38,47,28,41,42,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,d^b=di,e^c=eh,f^c=fh

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, 17719);