#10336785

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

4189 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)(17,18,22,21)(19,25,27,23)(20,28,30,24)(26,31,29,32)(33,34,40,43)(35,44,38,41)(36,39,37,42)(45,46,47,48);(1,3)(2,8)(4,5)(6,9)(7,12)(10,11)(13,14)(15,16)(17,19,22,27)(18,23,21,25)(20,29,30,26)(24,32,28,31)(33,35,40,38)(34,41,43,44)(36,47,37,45)(39,46,42,48);(2,9)(3,12)(5,13)(10,15)(18,21)(23,25)(24,28)(31,32)(33,36,40,37)(34,42,43,39)(35,45,38,47)(41,46,44,48);(1,4)(2,10)(3,13)(5,12)(6,11)(7,14)(8,16)(9,15)(17,20,22,30)(18,24,21,28)(19,26,27,29)(23,31,25,32)(33,37,40,36)(34,39,43,42)(35,45,38,47)(41,46,44,48);(1,5,7,13)(2,11,9,16)(3,4,12,14)(6,15,8,10)(17,21,22,18)(19,25,27,23)(20,28,30,24)(26,32,29,31)(33,38,40,35)(34,41,43,44)(36,47,37,45)(39,48,42,46);(1,6)(2,3)(4,11)(5,15)(7,8)(9,12)(10,13)(14,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,31)(28,32)(29,30)(33,39)(34,36)(35,46)(37,43)(38,48)(40,42)(41,47)(44,45)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=g,e^2=h,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, 10243981);