#10349102

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

4138 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=ghi,d^2=h,e^2=h,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, 10256298);