#10171865

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

3850 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)(17,18,22,23)(19,20,25,24)(21,29,31,26)(27,30,32,28)(33,34,38,41)(35,40,44,39)(36,37,47,45)(42,48,46,43);(1,2)(3,8)(4,9)(5,10)(6,7)(11,13)(12,15)(14,16)(18,23)(20,24)(26,29)(28,30)(33,34,38,41)(35,40,44,39)(36,45,47,37)(42,43,46,48);(17,19,22,25)(18,24,23,20)(21,27,31,32)(26,30,29,28)(33,35,38,44)(34,39,41,40)(36,42,47,46)(37,43,45,48);(1,3,6,10)(2,8,7,5)(4,12,14,11)(9,15,16,13)(17,19)(18,24)(20,23)(21,27)(22,25)(26,30)(28,29)(31,32)(33,35)(34,40)(36,46)(37,43)(38,44)(39,41)(42,47)(45,48);(1,4,6,14)(2,9,7,16)(3,11,10,12)(5,15,8,13)(17,20,22,24)(18,25,23,19)(21,30,31,28)(26,27,29,32)(33,36,38,47)(34,37,41,45)(35,42,44,46)(39,43,40,48);(1,5,6,8)(2,10,7,3)(4,13,14,15)(9,11,16,12)(17,21,22,31)(18,26,23,29)(19,27,25,32)(20,28,24,30)(33,37)(34,36)(35,43)(38,45)(39,42)(40,46)(41,47)(44,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=ghi,e^2=gi,f^2=g,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, 10079061);