#8371247

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

3842 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=h,d^2=ghi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8278443);