#8256611

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	#55941	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3918 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=g,d^2=gh,e^2=hi,f^2=ghi,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, 8163807);