#10196059

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3914 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10103255);