#10195584

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	#55336	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3963 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=i,e^2=gh,f^2=ghi,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, 10102780);