#8418427

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	#42504	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3932 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8325623);