#8119921

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	#47670	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8027117);