#8371566

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	#34589	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3944 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=g,e^2=g,f^2=gi,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, 8278762);