#10246976

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	#45741	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3995 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=gi,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, 10154172);