#8650387

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	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4020 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gi,d^2=ghi,e^2=g,f^2=gi,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, 8557583);