#8291639

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	#2539	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4036 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

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, 8198835);