#10247208

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	#2668	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3856 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:

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

Polycyclic presentation:

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

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, 10154404);