#8741697

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

3971 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)(3,7,13,8)(4,5,9,15)(11,12,16,14)(17,18)(19,25)(20,24)(21,26)(22,27)(23,30)(28,29)(31,32)(33,34)(35,43)(36,37)(38,40)(39,42)(41,47)(44,45)(46,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,10)(9,16)(14,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,27)(26,32)(33,35,39,38)(34,40,42,43)(36,46,41,44)(37,45,47,48);(1,3,6,13)(2,7,10,8)(4,11,9,16)(5,12,15,14)(17,20,22,26)(18,24,27,21)(19,28,30,32)(23,31,25,29)(33,35,39,38)(34,40,42,43)(36,46,41,44)(37,45,47,48);(1,3)(2,8)(4,11)(5,14)(6,13)(7,10)(9,16)(12,15)(17,20)(18,24)(19,28)(21,27)(22,26)(23,31)(25,29)(30,32)(33,36,39,41)(34,37,42,47)(35,44,38,46)(40,48,43,45);(1,4)(2,5)(3,11)(6,9)(7,12)(8,14)(10,15)(13,16)(17,19)(18,25)(20,28)(21,31)(22,30)(23,27)(24,29)(26,32)(33,37)(34,41)(35,45)(36,42)(38,48)(39,47)(40,44)(43,46);(1,5)(2,9)(3,12)(4,10)(6,15)(7,16)(8,11)(13,14)(17,21,22,24)(18,26,27,20)(19,29,30,31)(23,32,25,28)(33,38,39,35)(34,40,42,43)(36,44,41,46)(37,45,47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8648893);