#8260667

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	#57268	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3994 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gi,d^2=h,e^2=ghi,f^2=hi,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, 8167863);