#10166298

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	#56403	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4017 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=hi,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10073494);