#8944440

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4090 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)(3,9)(4,5)(6,14)(7,8)(10,15)(11,13)(12,16)(17,18)(19,20)(21,29)(22,23)(24,30)(25,26)(27,32)(28,31)(33,34,39,38)(35,41,36,40)(37,42,46,45)(43,48,44,47);(1,3,7,4)(2,5,8,9)(6,11,15,12)(10,13,14,16)(17,19)(18,20)(21,27)(22,25)(23,26)(24,28)(29,32)(30,31)(33,35,39,36)(34,40,38,41)(37,43,46,44)(42,47,45,48);(2,8)(5,9)(6,15)(11,12)(18,23)(19,25)(21,30)(28,32)(33,35,39,36)(34,41,38,40)(37,44,46,43)(42,47,45,48);(1,4,7,3)(2,9,8,5)(6,12,15,11)(10,16,14,13)(17,20)(18,19)(21,28)(22,26)(23,25)(24,27)(29,31)(30,32)(33,36,39,35)(34,41,38,40)(37,43,46,44)(42,47,45,48);(1,5,7,9)(2,4,8,3)(6,13,15,16)(10,12,14,11)(17,18)(19,26)(20,25)(21,24)(22,23)(27,32)(28,31)(29,30)(33,37,39,46)(34,42,38,45)(35,43,36,44)(40,47,41,48);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,14)(9,16)(17,21,22,30)(18,24,23,29)(19,27,25,31)(20,28,26,32)(33,38,39,34)(35,41,36,40)(37,45,46,42)(43,48,44,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8851636);