#10166258

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3862 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)(3,8)(4,5)(6,14)(7,9)(10,16)(11,13)(12,15)(18,23)(19,26)(25,32)(27,28)(33,34,37,40)(35,41,43,38)(36,39,46,44)(42,48,45,47);(1,3,7,4)(2,8,9,5)(6,11,16,12)(10,15,14,13)(17,18)(19,24)(20,25)(21,26)(22,23)(27,29)(28,31)(30,32)(33,34)(35,38)(36,44)(37,40)(39,46)(41,43)(42,48)(45,47);(2,9)(3,4)(10,14)(11,12)(33,35,37,43)(34,38,40,41)(36,42,46,45)(39,47,44,48);(1,4,7,3)(2,5,9,8)(6,12,16,11)(10,13,14,15)(17,19,22,26)(18,24,23,21)(20,28,30,27)(25,31,32,29)(33,35)(34,38)(36,45)(37,43)(39,48)(40,41)(42,46)(44,47);(1,5)(2,4)(3,9)(6,15)(7,8)(10,11)(12,14)(13,16)(17,20,22,30)(18,25,23,32)(19,27,26,28)(21,31,24,29)(33,36)(34,39)(35,42)(37,46)(38,47)(40,44)(41,48)(43,45);(1,6,7,16)(2,10,9,14)(3,11,4,12)(5,13,8,15)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,27,32,28)(33,34)(35,38)(36,44)(37,40)(39,46)(41,43)(42,48)(45,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,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, 10073454);