#10246878

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

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, 10154074);