#10038472

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	#38509	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4063 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9945668);