#10328312

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

4127 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,d^2=h,e^2=gh,f^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, 10235508);