#10166277

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3937 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=hi,d^2=gh,e^2=ghi,f^2=h,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, 10073473);