#8696757

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	#45833	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4090 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8603953);