#8621459

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3960 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8528655);