#8755092

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3847 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662288);