#10086333

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3869 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)(3,6)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(17,18,22,20)(19,24,27,23)(21,28,29,25)(26,30,32,31)(34,39)(35,41)(37,46)(43,47);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,18)(19,23)(20,22)(21,25)(24,27)(26,30)(28,29)(31,32)(34,39)(35,41)(36,45)(44,48);(2,7)(6,13)(8,11)(14,15)(17,19,22,27)(18,23,20,24)(21,26,29,32)(25,30,28,31)(33,34,38,39)(35,42,41,40)(36,46,45,37)(43,44,47,48);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,35,38,41)(34,40,39,42)(36,43,45,47)(37,44,46,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,20,22,18)(19,23,27,24)(21,25,29,28)(26,31,32,30)(33,36,38,45)(34,37,39,46)(35,43,41,47)(40,44,42,48);(1,4,5,12)(2,8,7,11)(3,9,10,16)(6,14,13,15)(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)(33,37)(34,36)(35,44)(38,46)(39,45)(40,43)(41,48)(42,47)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=i,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=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, 9993529);