#8097071

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

4155 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)(3,8)(4,12)(5,9)(6,10)(7,14)(11,13)(15,16)(17,18,23,24)(19,25,20,21)(22,26,31,32)(27,30,28,29)(33,34)(35,42)(36,40)(37,39)(38,41)(43,48)(44,47)(45,46);(1,3)(2,5)(4,11)(6,9)(7,13)(8,10)(12,15)(14,16)(18,24)(19,20)(26,32)(27,28)(36,44)(40,47)(43,45)(46,48);(17,19,23,20)(18,21,24,25)(22,27,31,28)(26,29,32,30)(33,35,38,37)(34,39,41,42)(36,43,44,45)(40,46,47,48);(1,4,6,14)(2,7,10,12)(3,11,9,16)(5,13,8,15)(17,20,23,19)(18,25,24,21)(22,28,31,27)(26,30,32,29)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,47)(42,48);(1,3)(2,8)(4,11)(5,10)(6,9)(7,15)(12,13)(14,16)(17,21,23,25)(18,19,24,20)(22,30,31,29)(26,28,32,27)(33,37,38,35)(34,39,41,42)(36,45,44,43)(40,46,47,48);(1,5,6,8)(2,9,10,3)(4,13,14,15)(7,16,12,11)(17,22,23,31)(18,26,24,32)(19,27,20,28)(21,29,25,30)(33,34)(35,39)(36,40)(37,42)(38,41)(43,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004267);