#8360009

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	#55689	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4095 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=h,d^2=ghi,f^2=g,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, 8267205);