#10422603

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	#41268	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3967 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,d^2=h,e^2=gh,f^2=gh,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, 10329799);