#8533462

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3739 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=hi,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440658);