#8942661

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

4088 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,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,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, 8849857);