#9054439

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
prime radical	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3775 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=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, 8961635);