#237166

order	512
nilpotency class	3
isoclinism	isoclinic class of #237155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1189 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,d^b=dhi,e^b=eh,f^b=fhi,g^b=gh,d^c=dh,e^c=ei,f^c=fhi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144362);