#9850861

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

3891 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,d^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,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, 9758057);