#8418641

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

3790 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,d^2=h,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, 8325837);