#110203

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	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

634 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=h,f^2=hi,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, 17399);