#8256606

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	#54289	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3911 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)(3,10)(4,14)(5,9)(6,7)(8,11)(12,13)(15,16)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(33,34,39,38)(35,42,41,37)(36,40,44,46)(43,48,47,45);(2,7)(3,11)(4,9)(13,16)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(36,44)(40,46)(43,47)(45,48);(2,7)(5,14)(8,10)(13,16)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,35)(34,37)(36,43)(38,42)(39,41)(40,45)(44,47)(46,48);(1,3)(2,8)(4,12)(5,13)(6,11)(7,10)(9,15)(14,16)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,36,39,44)(34,40,38,46)(35,43,41,47)(37,45,42,48);(1,4,6,9)(2,5,7,14)(3,12,11,15)(8,13,10,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,37,39,42)(34,41,38,35)(36,45,44,48)(40,47,46,43);(1,5)(2,9)(3,13)(4,7)(6,14)(8,15)(10,12)(11,16)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,38,39,34)(35,42,41,37)(36,46,44,40)(43,48,47,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,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, 8163802);