#8174313

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

4207 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)(18,23)(21,29)(24,25)(27,28)(33,34,38,41)(35,42,39,36)(37,45,46,40)(43,44,47,48);(1,3,7,13)(2,8,11,16)(4,12,9,6)(5,10,14,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,46)(47,48);(1,4,7,9)(2,5,11,14)(3,6,13,12)(8,15,16,10)(17,19,22,20)(18,24,23,25)(21,27,29,28)(26,31,30,32)(33,35,38,39)(34,36,41,42)(37,43,46,47)(40,44,45,48);(1,4)(2,5)(3,12)(6,13)(7,9)(8,10)(11,14)(15,16)(17,18)(19,25)(20,24)(21,26)(22,23)(27,32)(28,31)(29,30)(33,36,38,42)(34,39,41,35)(37,44,46,48)(40,47,45,43);(1,5)(2,9)(3,10)(4,11)(6,8)(7,14)(12,16)(13,15)(17,20,22,19)(18,25,23,24)(21,28,29,27)(26,32,30,31)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,46)(47,48);(1,6,7,12)(2,10,11,15)(3,4,13,9)(5,8,14,16)(17,21,22,29)(18,26,23,30)(19,27,20,28)(24,31,25,32)(33,37,38,46)(34,40,41,45)(35,43,39,47)(36,44,42,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gi,d^2=h,e^2=gh,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, 8081509);