#10352490

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	#46534	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3962 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10259686);