#10246861

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3929 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10154057);