#10424690

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	#49751	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4032 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=g,e^2=g,f^2=h,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, 10331886);