#10451657

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3890 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=hi,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10358853);