#110214

order	512
nilpotency class	3
isoclinism	isoclinic class of #110191
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

575 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)(3,10,11,14)(4,6,12,8)(7,15,16,13)(17,18)(19,28,31,24)(20,32)(21,37)(22,27)(23,26)(25,35)(29,44,33,46)(30,40,38,41)(34,47)(36,43)(39,45,42,48);(2,6,9,8)(3,11)(7,10,16,14)(13,15)(17,19,22,30,23,31,21,38)(18,24,27,41,26,28,37,40)(20,33,36,48,34,29,35,45)(25,42,32,46,43,39,47,44);(1,3,12,15,5,11,4,13)(2,7,6,14,9,16,8,10)(17,20,22,36,23,34,21,35)(18,25,37,47,26,43,27,32)(19,29,30,45,31,33,38,48)(24,39,40,46,28,42,41,44)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efi,d^2=hi,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,c^b=ch,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17410);