#8260865

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	#37155	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3903 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)(3,4)(5,8)(6,13)(7,9)(10,15)(11,16)(12,14)(17,18,23,20)(19,24,22,26)(21,25,30,28)(27,31,29,32)(33,34,39,38)(35,42,37,40)(36,41,46,45)(43,48,44,47);(1,3,8,7)(2,9,5,4)(6,11,15,14)(10,16,13,12)(17,19,23,22)(18,24,20,26)(21,29,30,27)(25,32,28,31)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(1,4)(2,7)(3,5)(6,12)(8,9)(10,14)(11,13)(15,16)(17,20,23,18)(19,24,22,26)(21,28,30,25)(27,31,29,32)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,5,8,2)(3,9,7,4)(6,13,15,10)(11,16,14,12)(33,37,39,35)(34,42,38,40)(36,44,46,43)(41,48,45,47);(1,6)(2,10)(3,11)(4,12)(5,13)(7,14)(8,15)(9,16)(17,21,23,30)(18,25,20,28)(19,27,22,29)(24,31,26,32);(1,7,8,3)(2,9,5,4)(6,14,15,11)(10,16,13,12)(17,22,23,19)(18,26,20,24)(21,29,30,27)(25,32,28,31)(33,38,39,34)(35,42,37,40)(36,45,46,41)(43,48,44,47)

Polycyclic presentation:

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

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, 8168061);