#10247187

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	#45868	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3937 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)(4,7)(5,10)(8,14)(9,11)(12,16)(13,15)(17,18,22,21)(19,25,27,23)(20,24,29,30)(26,32,28,31)(33,34)(35,44)(36,37)(38,48)(39,41)(40,45)(42,47)(43,46);(2,3)(4,11)(8,10)(12,13)(17,19,22,27)(18,23,21,25)(20,26,29,28)(24,31,30,32)(33,35,39,45)(34,40,41,44)(36,46,47,38)(37,48,42,43);(2,3)(7,9)(8,10)(15,16)(34,41)(35,45)(36,47)(43,48);(1,3,6,2)(4,7,11,9)(5,10,14,8)(12,15,13,16)(17,19)(18,23)(20,28)(21,25)(22,27)(24,32)(26,29)(30,31)(33,36)(34,42)(35,46)(37,41)(38,45)(39,47)(40,43)(44,48);(1,4,6,11)(2,7,3,9)(5,13,14,12)(8,16,10,15)(17,20)(18,24)(19,26)(21,30)(22,29)(23,31)(25,32)(27,28)(33,37,39,42)(34,36,41,47)(35,43,45,48)(38,44,46,40);(1,5,6,14)(2,8,3,10)(4,12,11,13)(7,15,9,16)(17,21,22,18)(19,25,27,23)(20,24,29,30)(26,31,28,32)(33,38,39,46)(34,43,41,48)(35,36,45,47)(37,44,42,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10154383);