#8942410

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

4232 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8849606);