#10325375

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

4110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=h,f^2=g,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, 10232571);