#10181242

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3907 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)(3,11,12,8)(4,9,13,16)(5,14,15,10)(34,40)(36,46)(41,43)(44,47);(2,7)(4,13)(8,11)(10,14)(17,18,23,21)(19,26,20,24)(22,25,30,29)(27,32,28,31)(33,34,39,40)(35,41,37,43)(36,42,46,38)(44,48,47,45);(1,3,6,12)(2,8,7,11)(4,10,13,14)(5,9,15,16)(17,19,23,20)(18,24,21,26)(22,27,30,28)(25,31,29,32)(33,35,39,37)(34,41,40,43)(36,44,46,47)(38,45,42,48);(1,3)(2,8)(4,14)(5,9)(6,12)(7,11)(10,13)(15,16)(17,20,23,19)(18,24,21,26)(22,28,30,27)(25,31,29,32)(33,36)(34,42)(35,44)(37,47)(38,40)(39,46)(41,48)(43,45);(1,4)(2,9)(3,10)(5,11)(6,13)(7,16)(8,15)(12,14)(17,21,23,18)(19,26,20,24)(22,25,30,29)(27,31,28,32)(33,37,39,35)(34,43,40,41)(36,44,46,47)(38,45,42,48);(1,5,6,15)(2,10,7,14)(3,9,12,16)(4,8,13,11)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,38,39,42)(34,36,40,46)(35,45,37,48)(41,44,43,47)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gi,d^2=i,e^2=ghi,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, 10088438);