#8926583

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	#47446	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3897 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833779);