#8908215

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

4099 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8815411);