#8460686

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3946 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)(3,11)(4,15)(5,16)(6,10)(7,14)(8,13)(9,12)(18,24)(19,20)(26,30)(27,29)(33,34,39,43)(35,38,42,47)(36,46,45,40)(37,41,44,48);(1,3)(2,7)(4,12)(5,13)(6,14)(8,9)(10,11)(15,16)(17,18,23,24)(19,25,20,28)(21,26,31,30)(22,27,32,29)(33,35,39,42)(34,38,43,47)(36,37,45,44)(40,48,46,41);(17,19,23,20)(18,25,24,28)(21,27,31,29)(22,26,32,30)(33,36,39,45)(34,40,43,46)(35,44,42,37)(38,41,47,48);(1,4,6,13)(2,8,10,15)(3,12,14,5)(7,9,11,16)(17,20,23,19)(18,25,24,28)(21,29,31,27)(22,26,32,30)(33,35)(34,38)(36,44)(37,45)(39,42)(40,41)(43,47)(46,48);(1,5,6,12)(2,9,10,16)(3,13,14,4)(7,8,11,15)(17,21,23,31)(18,26,24,30)(19,27,20,29)(22,25,32,28)(33,37,39,44)(34,41,43,48)(35,45,42,36)(38,40,47,46);(1,2)(3,7)(4,15)(5,9)(6,10)(8,13)(11,14)(12,16)(17,22,23,32)(18,27,24,29)(19,26,20,30)(21,25,31,28)(33,38)(34,42)(35,43)(36,41)(37,40)(39,47)(44,46)(45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=g,e^2=gi,f^2=hi,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, 8367882);