#10420118

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	#41140	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4000 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gh,d^2=h,f^2=g,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, 10327314);