#8131193

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	#50087	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4200 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=ghi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8038389);