#10147866

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3850 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)(3,9)(4,5)(6,10)(7,13)(8,14)(11,15)(12,16)(17,18,22,25)(19,23,29,32)(20,21,30,24)(26,31,27,28)(33,34)(35,40)(36,43)(37,39)(38,45)(41,47)(42,44)(46,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,19,22,29)(18,23,25,32)(20,27,30,26)(21,28,24,31)(33,35,39,43)(34,36,37,40)(38,42,47,46)(41,44,45,48);(17,20,22,30)(18,24,25,21)(19,26,29,27)(23,28,32,31)(33,36,39,40)(34,35,37,43)(38,44,47,48)(41,42,45,46);(1,4,6,14)(2,8,10,5)(3,11,13,16)(7,15,9,12)(17,20)(18,21)(19,27)(22,30)(23,28)(24,25)(26,29)(31,32)(33,37,39,34)(35,40,43,36)(38,45,47,41)(42,48,46,44);(1,5,6,8)(2,4,10,14)(3,12,13,15)(7,11,9,16)(17,19)(18,23)(20,26)(21,31)(22,29)(24,28)(25,32)(27,30)(33,35)(34,40)(36,37)(38,46)(39,43)(41,44)(42,47)(45,48);(1,3)(2,9)(4,11)(5,15)(6,13)(7,10)(8,12)(14,16)(17,21)(18,20)(19,28)(22,24)(23,27)(25,30)(26,32)(29,31)(33,38)(34,41)(35,42)(36,44)(37,45)(39,47)(40,48)(43,46)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10055062);