#110210

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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

569 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=h,f^2=hi,g^2=h,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, 17406);