#8479565

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

3893 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,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, 8386761);