#8618019

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	#40930	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4038 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)(3,10)(4,12)(5,7)(6,8)(9,14)(11,15)(13,16)(33,34)(35,37)(36,46)(38,41)(39,42)(40,44)(43,45)(47,48);(1,3,7,6)(2,8,5,10)(4,13,14,11)(9,16,12,15)(17,18,22,23)(19,24,28,30)(20,27,29,25)(21,26,31,32)(33,35,39,40)(34,37,42,44)(36,47,41,43)(38,45,46,48);(2,5)(8,10)(9,12)(15,16)(18,23)(20,29)(24,30)(26,32)(33,36,39,41)(34,38,42,46)(35,43,40,47)(37,48,44,45);(1,4)(2,9)(3,11)(5,12)(6,13)(7,14)(8,15)(10,16)(17,19)(18,24)(20,26)(21,25)(22,28)(23,30)(27,31)(29,32)(33,34)(35,44)(36,38)(37,40)(39,42)(41,46)(43,45)(47,48);(1,5,7,2)(3,8,6,10)(4,12,14,9)(11,15,13,16)(17,20,22,29)(18,25,23,27)(19,26,28,32)(21,30,31,24)(33,37,39,44)(34,40,42,35)(36,45,41,48)(38,43,46,47);(1,6,7,3)(2,8,5,10)(4,11,14,13)(9,16,12,15)(17,21)(18,26)(19,27)(20,30)(22,31)(23,32)(24,29)(25,28)(33,38)(34,41)(35,45)(36,42)(37,43)(39,46)(40,48)(44,47)

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8525215);