#10451621

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

3885 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=hi,e^2=gi,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, 10358817);