#8115584

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	#39818	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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)(3,9)(4,11)(5,13)(6,7)(8,14)(10,12)(15,16)(21,29)(25,30)(26,31)(27,32)(33,34)(35,41)(36,40)(37,45)(38,48)(39,42)(43,46)(44,47);(1,3,6,11)(2,4,7,9)(5,10,14,16)(8,12,13,15)(17,18)(19,20)(21,25)(22,23)(24,28)(26,27)(29,30)(31,32)(33,35,39,45)(34,37,42,41)(36,43,38,44)(40,47,48,46);(2,7)(4,9)(8,13)(12,15)(18,23)(19,24)(25,30)(26,31)(33,36,39,38)(34,40,42,48)(35,43,45,44)(37,47,41,46);(1,4)(2,3)(5,12)(6,9)(7,11)(8,10)(13,16)(14,15)(17,19,22,24)(18,20,23,28)(21,26,29,31)(25,27,30,32)(33,37,39,41)(34,35,42,45)(36,46,38,47)(40,44,48,43);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,15)(14,16)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,30)(29,32)(33,35)(34,41)(36,43)(37,42)(38,44)(39,45)(40,46)(47,48);(1,5,6,14)(2,8,7,13)(3,10,11,16)(4,12,9,15)(17,21,22,29)(18,25,23,30)(19,26,24,31)(20,27,28,32)(33,38,39,36)(34,40,42,48)(35,44,45,43)(37,47,41,46)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022780);