#10451597

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	#48111	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3949 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gh,d^2=hi,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=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, 10358793);