#10042602

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	#2669	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3974 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)(3,5)(4,10)(6,8)(7,9)(11,14)(12,15)(13,16)(19,26)(24,27)(28,31)(29,32)(33,34,39,36)(35,42,37,40)(38,41,46,44)(43,48,45,47);(1,3,7,6)(2,8,9,5)(4,12,13,11)(10,14,16,15)(17,18,22,23)(19,27,26,24)(20,21,25,30)(28,32,31,29)(33,35,39,37)(34,40,36,42)(38,43,46,45)(41,47,44,48);(2,9)(4,13)(5,8)(11,12)(18,23)(20,25)(24,27)(28,31)(33,34,39,36)(35,40,37,42)(38,41,46,44)(43,47,45,48);(1,4,7,13)(2,10,9,16)(3,11,6,12)(5,14,8,15)(17,19,22,26)(18,24,23,27)(20,28,25,31)(21,29,30,32)(33,36,39,34)(35,40,37,42)(38,44,46,41)(43,47,45,48);(1,5,7,8)(2,3,9,6)(4,14,13,15)(10,11,16,12)(17,20)(18,21)(19,28)(22,25)(23,30)(24,29)(26,31)(27,32)(33,37,39,35)(34,40,36,42)(38,43,46,45)(41,48,44,47);(1,6,7,3)(2,5,9,8)(4,12,13,11)(10,14,16,15)(17,21)(18,25)(19,29)(20,23)(22,30)(24,31)(26,32)(27,28)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9949798);