#10480908

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4016 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,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, 10388104);