#10320351

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3856 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=ghi,d^2=ghi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10227547);