#8479563

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3882 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)(3,9)(4,12)(5,6)(7,8)(10,14)(11,13)(15,16)(20,28)(23,29)(27,31)(30,32)(33,34,40,44)(35,45,39,41)(36,47,46,42)(37,48,43,38);(2,8)(5,9)(10,12)(13,16)(17,18)(19,26)(20,29)(21,24)(22,25)(23,28)(27,30)(31,32)(33,35,40,39)(34,41,44,45)(36,43,46,37)(38,47,48,42);(1,3,7,6)(2,9,8,5)(4,11,14,15)(10,16,12,13)(17,19,22,24)(18,21,25,26)(20,27,28,31)(23,30,29,32)(33,36)(34,42)(35,37)(38,45)(39,43)(40,46)(41,48)(44,47);(1,4)(2,10)(3,11)(5,13)(6,15)(7,14)(8,12)(9,16)(17,18)(19,21)(20,23)(22,25)(24,26)(27,30)(28,29)(31,32)(33,37,40,43)(34,38,44,48)(35,46,39,36)(41,42,45,47);(1,5)(2,3)(4,13)(6,8)(7,9)(10,11)(12,15)(14,16)(17,20)(18,23)(19,27)(21,30)(22,28)(24,31)(25,29)(26,32)(33,38)(34,43)(35,47)(36,45)(37,44)(39,42)(40,48)(41,46);(1,6,7,3)(2,5,8,9)(4,11,14,15)(10,16,12,13)(17,21)(18,24)(19,25)(20,30)(22,26)(23,31)(27,29)(28,32)(33,39,40,35)(34,41,44,45)(36,43,46,37)(38,42,48,47)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8386759);