#8174374

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

3915 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:

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

Polycyclic presentation:

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

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, 8081570);