#9243754

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

4154 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)(3,5)(4,9)(6,7)(8,11)(10,13)(12,14)(15,16)(17,18)(19,22)(20,27)(21,24)(23,28)(25,29)(26,32)(30,31)(33,34,39,37)(35,40,45,47)(36,42,46,41)(38,48,44,43);(2,7)(5,8)(9,14)(13,15)(19,25)(22,29)(26,31)(30,32)(33,35,39,45)(34,40,37,47)(36,38,46,44)(41,43,42,48);(4,12)(9,14)(10,16)(13,15)(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32)(33,36,39,46)(34,41,37,42)(35,44,45,38)(40,48,47,43);(1,3)(2,8)(4,10)(5,7)(6,11)(9,15)(12,16)(13,14)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,36)(34,42)(35,38)(37,41)(39,46)(40,48)(43,47)(44,45);(1,4,6,12)(2,9,7,14)(3,10,11,16)(5,13,8,15)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,37,39,34)(35,40,45,47)(36,41,46,42)(38,48,44,43);(1,5,6,8)(2,3,7,11)(4,13,12,15)(9,10,14,16)(17,20)(18,23)(19,26)(21,28)(22,30)(24,27)(25,31)(29,32)(33,38)(34,43)(35,36)(37,48)(39,44)(40,41)(42,47)(45,46)

Polycyclic presentation:

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

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, 9150950);