#8551308

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

4305 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,e^2=g,f^2=ghi,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, 8458504);