#8757268

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

4039 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)(3,11)(4,13)(5,15)(6,7)(8,12)(9,14)(10,16)(17,18,22,19)(20,25,28,23)(21,26,29,24)(27,30,32,31)(35,41)(38,42)(43,47)(46,48);(2,7)(5,16)(8,11)(9,13)(17,18,22,19)(20,23,28,25)(21,26,29,24)(27,31,32,30)(33,34)(35,42)(36,44)(37,40)(38,41)(39,45)(43,46)(47,48);(2,7)(3,12)(9,13)(10,15)(18,19)(23,25)(24,26)(30,31)(33,34,37,40)(35,38,41,42)(36,39,45,44)(43,46,47,48);(1,3)(2,8)(4,10)(5,13)(6,12)(7,11)(9,16)(14,15)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,34)(35,38)(36,39)(37,40)(41,42)(43,46)(44,45)(47,48);(1,4)(2,9)(3,10)(5,11)(6,14)(7,13)(8,16)(12,15)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,35,37,41)(34,38,40,42)(36,43,45,47)(39,46,44,48);(1,5,6,16)(2,10,7,15)(3,13,12,9)(4,11,14,8)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,36)(34,39)(35,43)(37,45)(38,46)(40,44)(41,47)(42,48)

Polycyclic presentation:

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

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, 8664464);