#10412053

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

4315 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)(3,4)(5,9)(6,10)(7,8)(11,12)(13,15)(14,16)(17,18,22,23)(19,20,27,24)(21,29,31,25)(26,32,30,28)(34,39)(35,41)(37,45)(44,47);(1,2,6,10)(3,7,8,4)(5,9,14,16)(11,15,13,12)(18,23)(20,24)(25,29)(28,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(17,19)(18,24)(20,23)(21,26)(22,27)(25,32)(28,29)(30,31)(33,35)(34,36)(37,43)(38,41)(39,40)(42,44)(45,48)(46,47);(1,3)(2,7)(4,10)(5,13)(6,8)(9,12)(11,14)(15,16)(17,19)(18,24)(20,23)(21,30)(22,27)(25,28)(26,31)(29,32)(33,35)(34,40)(36,39)(37,43)(38,41)(42,47)(44,46)(45,48);(1,4,6,7)(2,8,10,3)(5,12,14,15)(9,13,16,11)(17,20,22,24)(18,19,23,27)(21,28,31,32)(25,26,29,30)(33,36,38,40)(34,41,39,35)(37,44,45,47)(42,48,46,43);(1,5)(2,9)(3,11)(4,12)(6,14)(7,15)(8,13)(10,16)(17,21)(18,25)(19,26)(20,28)(22,31)(23,29)(24,32)(27,30)(33,37)(34,42)(35,43)(36,44)(38,45)(39,46)(40,47)(41,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=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, 10319249);