#8907962

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

4212 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)(4,6)(5,8)(7,10)(9,14)(12,16)(13,15)(17,18,22,24)(19,21,27,23)(20,28,29,25)(26,32,30,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,45)(44,48)(46,47);(1,3,7,5)(2,8,10,11)(4,12,14,15)(6,13,9,16)(17,19,22,27)(18,23,24,21)(20,26,29,30)(25,32,28,31)(33,35)(34,40)(36,44)(37,46)(38,45)(39,43)(41,48)(42,47);(4,14)(6,9)(12,15)(13,16)(18,24)(20,29)(21,23)(26,30)(34,38)(35,43)(36,46)(42,48);(1,4)(2,9)(3,12)(5,15)(6,10)(7,14)(8,16)(11,13)(17,20,22,29)(18,25,24,28)(19,26,27,30)(21,31,23,32)(33,36,39,46)(34,41,38,47)(35,44,43,37)(40,48,45,42);(1,5,7,3)(2,8,10,11)(4,12,14,15)(6,16,9,13)(17,19)(18,21)(20,30)(22,27)(23,24)(25,32)(26,29)(28,31)(33,37,39,44)(34,42,38,48)(35,36,43,46)(40,41,45,47);(1,6,7,9)(2,4,10,14)(3,13,5,16)(8,12,11,15)(17,21)(18,19)(20,31)(22,23)(24,27)(25,26)(28,30)(29,32)(33,38,39,34)(35,45,43,40)(36,47,46,41)(37,48,44,42)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=h,e^2=gh,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, 8815158);