#10336792

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

4286 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)(3,8)(4,10)(5,11)(6,12)(7,9)(13,16)(14,15)(17,18)(19,25)(20,22)(21,24)(23,27)(26,31)(28,29)(30,32)(34,40)(35,36)(38,48)(42,45);(1,3)(2,8)(4,5)(6,9)(7,12)(10,11)(13,14)(15,16)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(33,34,39,40)(35,41,36,44)(37,38,47,48)(42,46,45,43);(2,9)(3,12)(5,13)(10,15)(21,29)(24,28)(26,32)(30,31)(33,35,39,36)(34,41,40,44)(37,45,47,42)(38,43,48,46);(1,4)(2,10)(3,13)(5,12)(6,11)(7,14)(8,16)(9,15)(17,20,22,18)(19,23,27,25)(21,24,29,28)(26,31,32,30)(33,36,39,35)(34,41,40,44)(37,45,47,42)(38,46,48,43);(1,5,7,13)(2,11,9,16)(3,4,12,14)(6,15,8,10)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)(33,37)(34,38)(35,42)(36,45)(39,47)(40,48)(41,46)(43,44);(1,6)(2,3)(4,11)(5,15)(7,8)(9,12)(10,13)(14,16)(17,19)(18,25)(20,23)(21,26)(22,27)(24,31)(28,30)(29,32)(33,38)(34,37)(35,43)(36,46)(39,48)(40,47)(41,45)(42,44)

Polycyclic presentation:

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

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, 10243988);