#10247634

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

3992 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gh,d^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10154830);