#10216821

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

3912 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,e^2=ghi,f^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, 10124017);