#8094541

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

4128 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:

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

Polycyclic presentation:

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

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, 8001737);