#8425651

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	#2540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3945 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=i,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332847);