#10055406

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	#55553	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4125 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=g,e^2=i,f^2=g,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, 9962602);