#10349101

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

4166 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,16)(13,15)(17,18,22,20)(19,23,27,25)(21,28,29,24)(26,31,32,30)(33,34,40,43)(35,44,38,41)(36,39,37,42)(45,46,47,48);(2,7)(3,11)(5,14)(12,15)(17,18,22,20)(19,25,27,23)(21,28,29,24)(26,30,32,31)(33,35,40,38)(34,41,43,44)(36,47,37,45)(39,46,42,48);(2,7)(4,9)(8,10)(12,15)(18,20)(23,25)(24,28)(30,31)(33,36,40,37)(34,42,43,39)(35,45,38,47)(41,46,44,48);(1,3,6,11)(2,8,7,10)(4,12,9,15)(5,13,14,16)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(33,37,40,36)(34,39,43,42)(35,45,38,47)(41,46,44,48);(1,4,6,9)(2,5,7,14)(3,12,11,15)(8,13,10,16)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30)(33,38,40,35)(34,41,43,44)(36,47,37,45)(39,48,42,46);(1,5)(2,9)(3,13)(4,7)(6,14)(8,15)(10,12)(11,16)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)(33,39)(34,36)(35,46)(37,43)(38,48)(40,42)(41,47)(44,45)

Polycyclic presentation:

2^9|a^2=i,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, 10256297);