#8463673

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3965 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8370869);