#10216992

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3773 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^2=i,e^2=ghi,f^2=gi,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, 10124188);