#10352519

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

4095 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10259715);