#10412057

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

4294 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,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, 10319253);