#8739195

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	#52839	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3945 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=g,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8646391);